Airborne Wind Profiling Portable Radar System and Method

ABSTRACT

An airborne wind profiling portable radar (AWiPPR) system comprising a mobile airborne platform including one or more navigation units configured to produce navigation data including at least the position and orientation of the mobile airborne platform. A radar unit is mounted and positioned to the mobile airborne platform, the radar unit is configured to transmit a wide-band frequency modulated continuous wave radar signal in a downward direction from the mobile airborne platform towards the ground and configured to continuously receive a reflected signal from a plurality of clear air scatters (CAS) targets or volumetric targets and output radar data. An inertial measurement unit (IMU) in communication with the one or more navigation units and the radar unit is configured to receive the navigation data and determine the position and orientation of the radar at a specific point in time and output IMU data. A data acquisition unit in communication with the radar unit and the is configured to receive and time align radar data and the IMU data for each reflected signal from each of the plurality of CAS targets or volumetric targets to provide an antenna pointing direction for each received reflected signal. The data acquisition unit is configured to process the time aligned radar data and IMU data to determine a distance and a Doppler velocity of each of the plurality of CAS targets or volumetric targets, provide a range, a velocity, and an antenna pointing direction for each of the plurality of CAS targets or volumetric targets, and calculate a vector wind velocity using the range, the velocity, and the antenna pointing direction for each of the plurality of CAS targets or volumetric scatters targets, The data acquisition unit may be configured to further correct the range, the velocity, and/or the antenna pointing direction of each of the plurality of CAS targets or volumetric targets to accommodate for a motion shift in data produced by one or more on a relative motion and orientation of the mobile airborne platform, a Doppler spread in the range, the velocity and/or the antenna pointing direction, and a ground echo.

RELATED APPLICATIONS

This application claims benefit of and priority to U.S. ProvisionalApplication Ser. No. 62/589,650 tiled Nov. 22, 2017, under 35 U.S.C. §§119, 120, 363, 365, and 37 C.F.R. § 1.55 and § 1.78, which isincorporated herein by this reference.

FIELD OF THE INVENTION

This subject invention relates to an airborne wind profiling portableradar system and method.

BACKGROUND OF THE INVENTION

Wind profilers are Doppler radars that typically operate in the VHF(30-300 MHz) or UHF (300-1000 MHz) frequency bands. There are threeprimary types of radar wind profilers in operation in the U.S. today.The NOAA Profiler Network (NPN) profiler operates at a frequency of 404MHz. The second type of profiler that is used by NOAA research andoutside agencies is the 915-MHz boundary-layer profiler. The 404-MHzprofilers are more expensive to build and operate, but they provide thedeepest coverage of the atmosphere. The 915-MHz profilers are smallerand cheaper to build and operate, but they lack height coverage muchabove the boundary layer. A third type of profiler that operates at 449MHz (the so-called ¼-scale 449-MHz profilers) combines the best samplingattributes of the other two systems. See Table 1 below

TABLE 1 Physical, operating, and sampling characteristics of windprofilers. 404-MHz 915-MHz 449-MHz (NPN) (boundary layer) (quarter-scale) Antenna type Coaxial-colinear Flat rectangular Coaxial-colinearphased array microstrip patch phased array Antenna diameter (m) 13  2  6Beamwidth (deg.)  4 10 10 Peak transmit power (W) 6000  500  2000 Transmit pulse width (μs) 3.3^(p), 20^(p)   0.417*, 0.708*^(,p) 0.708*,2.833  Height coverage (m) 500**-16,000 120-4,000 180⁺-8,000 Verticalresolution (m) 320+, 900+  63, 106*  106, 212*^(,) + Temporal resolution(min) 60 60 60 *These settings reflect how the profilers were operatedduring typical deployments. Other degraded transmit and samplingresolutions are possible. ^(p)Pulse-coding was used in selectedoperating modes to boost signal power and increase altitude coverage(for more information on pulse coding, see Ghebrebrhan, 1990). ⁺Thisminimum detectable range has been achieved with the ¼-scale 449-MHzprofilers using a 0.7-μs pulse. **Signal attenuators prevent accurateradar reflectivity data below 1 km. +Increased vertical resolution ascompared to the transmit pulse length was accomplished by oversampling.

See also U.S. Pat. Nos. 7,109,913; 9,310,481; and 9,007,570,incorporated by this reference herein. However, the conventional windprofilers discussed above are ground based systems have large apertures(antenna diameters), large range cells (vertical resolution), largeblanking ranges (height coverage) and slow response (temporalresolution) and, therefore, cannot be used effectively in an airborneplatform to determine vector wind velocity as a function of altitudeabove the ground

BRIEF SUMMARY OF THE INVENTION

Featured is an airborne wind profiling portable radar (AWiPPR) systemcomprising a mobile airborne platform including one or more navigationunits configured to produce navigation data including at least theposition and orientation of the mobile airborne platform. A radar unitis mounted and positioned to the mobile airborne platform, the radarunit is configured to transmit a wide-band frequency modulatedcontinuous wave radar signal in a downward direction from the mobileairborne platform towards the ground and configured to continuouslyreceive a reflected signal from a plurality of clear air scatters (CAS)targets or volumetric targets and output radar data. An inertialmeasurement unit in communication with the one or more navigation unitsand the radar unit, is configured to receive the navigation data anddetermine the position and orientation of the radar at a specific pointin time and output IMU data. A data acquisition unit in communicationwith the radar unit and the IMU is configured to receive and time alignradar data and the NU data for each reflected signal from each of theplurality of CAS targets or volumetric targets to provide an antennapointing direction for each received reflected signal. The dataacquisition unit is configured to process the time aligned radar dataand IMU data to determine a distance and a Doppler velocity of each ofthe plurality of CAS targets or volumetric targets, provide a range, avelocity, and an antenna pointing direction for each of the plurality ofCAS targets or volumetric targets, and calculate a vector wind velocityusing the range, the velocity, and the antenna pointing direction foreach of the plurality of CAS targets or volumetric scatters targets. Thedata acquisition unit may be configured to further correct the range,the velocity, and/or the antenna pointing direction of each of theplurality of CAS targets or volumetric targets to accommodate for amotion shift in data produced by one or more of: a relative motion andorientation of the mobile airborne platform, a Doppler spread in therange, the velocity and/or the antenna pointing direction, and a groundecho.

In one embodiment, a motion shift in the range, the velocity, and theantenna pointing direction may not include a Doppler wrap and whereinthe navigation unit may be configured to generate a navigationcorrection for each reflected signal by adding a speed of the mobileairborne platform provided by the one or more navigation units to thedetermined Doppler velocity for each of the plurality of CAS targets orvolumetric targets and rotating the range, the velocity, and the antennapointing direction into a coordinate system centered beneath the mobileairborne platform. The data acquisition unit may be responsive tosparseness of the plurality of CAS targets, shifts in the position ofthe mobile airborne platform over a predetermined measurement window,and navigation correction applied to a set of reflected signals from theplurality of CAS targets or volumetric targets and the data acquisitionunit may be configured to infer a Doppler field vector for each of theplurality of CAS targets or volumetric targets as a set of three coupledcubic splines derived from the measured Doppler velocity data for theplurality of CAS targets or volumetric using a non-parametric functionestimation. The data acquisition unit may be configured to generate avector wind field from the set of three coupled cubic splines by:representing the second derivative of the cubic splines as a piecewisecontinuous linear function (f(z)), integrating the function twice toyield a cubic polynomial producing a plurality of pivot points of thecubic splines, wherein the function (f(z)) must pass through the pivotpoints and be zero at the first and last pivot points such that thecubic splines are natural splines, determining a plurality of unknownspline ordinate points from the altitude and velocity data obtained bythe data acquisition unit, wherein a minimum spline abscissa value isequal to the minimum altitude and velocity data values, and wherein amaximum spline abscissa value is equal to the maximum altitude andvelocity data values, wherein the abscissa of the altitude and velocitydata lies in an abscissa interval of the cubic splines, and wherein theordinate points represent the velocity of the unknown wind field,wherein such the abscissa intervals are determined by ensuring that allabscissa intervals contain equal amounts of information, and whereinsuch that the relationship between the observed velocity data and thecubic splines is given by: V_(N) _(d) _(xi)=A_(N) _(d) _(×3N) where V isthe vector of the obtained velocity data, N is the number of datapoints, f is the vector of a set of cubic spline coefficients, A is aninformation matrix, and Af is a cubic spline model. The data acquisitionunit may fit the cubic spline model Af to the obtained velocity data Vusing a least-squares technique. The data acquisition unit may minimizea difference between the obtained velocity data V and the cubic splinemodel Af by obtaining a maximum likelihood estimate off The dataacquisition unit may determine a required minimum slant distance of theradar unit relative to the ground from the reflected signal that yieldsa maximum allowable return signal into the radar unit before theperformance of the radar unit is reduced to saturation or compression.The data acquisition unit may be configured to determine the requiredincidence angle using a Beckman and Spizzichino model. A pointing angleof the radar unit relative to the mobile airborne platform may beadjustable and the radar unit adjusts the pointing angle of the radarunit based on the determined required incidence angle. The pointingangle of the radar unit may be pointed at an angle relative to a normalto the ground of greater than about 0° and less than about 90°. The dataacquisition unit may be further configured to estimate the vector windvelocity by: selecting a plurality of measurements containing a CAStargets or volumetric target and determining a slant distance andDoppler velocity of a ground echo from each, performing the requiredcoordinate transformations such that the range and Doppler velocity ofthe ground echo are at zero range and velocity, extracting a slantdistance and Doppler wind velocity for each of the CAS targets orvolumetric targets in the plurality of measurements above a fixedsignal-to-noise threshold, and converting the slant distance to analtitude above ground level using the navigation data from the one ormore navigation units. The data acquisition unit may be furtherconfigured to minimize the chi-square sum between the measured windvector velocities and the estimated wind vector velocities by a gradientsearch technique. The radar unit may transmit with a sweep widthconfigured to match the backscattering characteristics of the pluralityof CAS targets or volumetric targets. The sweep widths may range fromabout 6 MHz to about 200 MHz. The radar unit may transmit in a waveformselected from one or more of linear frequency modulated (FM) waveform, aphase coded waveform, or non-linear FM waveform. The radar unit may beconfigured to transmit at a carrier frequency in the Ka band. The radarunit may be configured to convert the wide-band frequency modulatedcontinuous wave radar signal to a Ka band and filter and amplify the Kaband signal prior to transmission thereof The radar unit may beconfigured to receive the reflected signal from each of the plurality ofCAS targets or volumetric targets, amplify the received signal,down-convert the received signal to a baseband received signal, andfilter and amplify the received signal. The down conversion may behomodyne single side band. The down-conversion may be homodyne and isdual side band. The radar unit may include one or more antennas.

Also featured is a method of determining a vector wind velocity anddirection as a function of altitude above the ground on a mobileairborne platform. The method comprising providing navigation dataincluding at least positioning and orientation of the mobile airborneplatform. A wide band frequency modulated continuous wave radar signalis transmitted in a downward direction from the mobile airborne platformtowards the ground. A reflected signal from each of a plurality of cleanair scatter (CAS) targets or volumetric targets is continuously receivedand radar data is output. The position and orientation of a radar unitmounted and positioned on the mobile airborne platform is determined ata specific point in time and position and orientation data are output.The radar data and the position and orientation data for each reflectedsignal from each of the plurality of CAS targets or volumetric targetsare time aligned to provide an antenna pointing direction for each ofthe plurality of CAS targets or volumetric targets. The timed alignedradar data and position and orientation data are processed to determinea distance and Doppler velocity for each of the plurality of CAS targetsor volumetric targets and provide a range, a velocity, and an antennapointing direction for each of the plurality of CAS targets orvolumetric targets and a vector wind velocity is calculated using therange, the velocity, and the antenna pointing direction for each of theplurality of CAS targets or volumetric targets. The range, the velocity,and/or the antenna pointing direction of each of the plurality of CAStargets or volumetric targets is further corrected to accommodate for amotion shift in the data produced by one or more of: a relative motionin orientation of the mobile airborne platform, a Doppler spread in therange, the velocity, and/or the antenna pointing direction and a groundecho.

In one embodiment, a shift in the range, the velocity, and the antennapointing direction may not include a Doppler wrap and a navigationcorrection for each reflected signal is generated by adding a speed ofthe mobile airborne platform to the determined Doppler velocity for eachof the plurality of CAS targets or volumetric targets and the range, thevelocity, and the antenna pointing direction is rotated into acoordinate system centered beneath the airborne platform. The method mayinclude detecting sparseness of the plurality of CAS targets, shifts inposition of the mobile airborne platform over a predeterminedmeasurement window and navigation correction applied to set of reflectedsignals from the plurality of CAS targets and inferring a Doppler fieldvector for each of the plurality of CAS targets or volumetric targets asa set of three cubic splines derived from the measured Doppler velocitydata for the plurality of CAS targets or volumetric scatters targetsusing a non-parameteric function estimation. The method may includegenerating a vector field from each of a set of cubic splines by:representing the second derivative of the cubic spines as a piecewisecontinuous linear function (f(z)), integrating the function twice toyield a cubic polynomial producing a plurality of pivot points of thecubic splines, wherein the function (f(z)) must pass through the pivotpoints and be zero at the first and last pivot points such that thecubic splines are natural splines, determining a plurality of unknownspine ordinate points from the altitude and velocity data obtained bythe data acquisition unit, wherein a minimum spline abscissa value isequal to the minimum altitude and velocity data values, and wherein amaximum spline abscissa value is equal to the maximum altitude andvelocity data values, wherein the abscissa of the altitude and velocitydata lies in an abscissa interval of the cubic spines, and wherein theordinate points represent the velocity of the unknown wind field,wherein such the abscissa intervals may be determined by ensuring thatall abscissa intervals contain equal amounts of information. Whereinsuch that the relationship between the observed velocity data and thecubic spines is given by: V_(N) _(d) _(×i)=A_(N) _(d) _(×3N)f_(SN),where V is the vector of the obtained velocity data, N is the number ofdata points, f is the vector of a set of cubic spline coefficients, A isan information matrix, and Af is a cubic spline model. The method mayinclude fitting the cubic spline model of Af to obtain the velocity datausing an at least squares technique. The method may include minimizingthe difference between the obtained velocity data V and the cubic splinemodel Af by obtaining a maximum likelihood estimate off. The method mayinclude determining a required minimum slant distance of a radar unitdisposed on the mobile airborne unit relative to the ground from thereflected signal that yields a maximum allowable signal return before aperformance of the radar unit is reduced to saturation or compression.The method may include determining the required incidence angle using aBeckman and Spizzichino model. The method may include providing apointing angle of the radar unit relative to the mobile airborneplatform that is adjustable and the radar unit adjusts a pointing angleof the radar unit based on the determined required incidence angle. Thepointing angle of the radar unit may be pointed at an angle relative toa normal to the ground of greater than about 0° and less than about 90°.The method may include estimating the wind velocity vector by: selectinga plurality of measurements containing a CAS target or volumetricscatter target and determining a slant distance and Doppler velocity ofa around echo from each, performing the required coordinatetransformations such that the range and Doppler velocity of the groundecho are at zero range and velocity and extracting a slant distance andDoppler wind velocity for each of the CAS targets in the plurality ofmeasurements above a fixed signal-to-noise threshold, converting theslant distance to an altitude above ground level using the navigationdata from the one or more navigation units. The method may includeminimizing the Chi-square sum between the measured wind vector velocityand the estimated wind vector velocity by a gradient search technique.The wide-band frequency modulator continuous wave radar signal maytransmit with a sweep width configured to match the back-scatteringcharacteristics of the CAS targets or volumetric targets. The sweepwidths may be in the range of about 6 MHz to about 200 MHz. Thewide-band frequency band modulator continuous wave radar signal mayinclude one or more of a linear frequency modulated (FM) waveform, aphase coded waveform, or non-linear FM waveform. The wide-band frequencymodulator continuous wave radar signal may transmit a carrier frequencyin the Ka band. The method may include converting the wide-bandfrequency modulator continuous wave radar signal to a Ka band, filteringand amplifying the Ka band prior to transmission thereof. The method mayinclude receiving the reflected signal from each of the plurality of CAStargets or volumetric scatters targets, amplifying the received signal,down converting the received signal to a base band received signal andfiltering and amplifying the received signal. The down conversion may bea homodyne single side band. The down conversion may be a homodyne andis a dual side band.

The subject invention, however, in other embodiments, need not achieveall these objectives and the claims hereof should not be limited tostructures or methods capable of achieving these objectives.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

Other objects, features and advantages will occur to those skilled inthe art from the following description of a preferred embodiment and theaccompanying drawings, in which:

FIG. 1 is a schematic side view showing one example of the primarycomponents of one embodiment of a airborne wind profiling portable radar(AWiPPR) system of this invention;

FIG. 2 is a three-dimensional view showing one example of a prototype ofthe AWiPPR system 10 shown in FIG. 1 which is mounted in the mobileairborne platform;

FIG. 3 is a three-dimensional view showing one example of the prototypeof the AWiPPR system 10 shown in FIG. 2 mounted in the mobile airborneplatform shown in FIG. 2;

FIG. 4 is a schematic block diagram showing in further detail theprimary components of the AWiPPR system 10 shown ire FIGS. 1-3;

FIG. 5 is a block diagram showing the primary steps of one embodiment ofthe method for determining vector wind velocity on a mobile airborneplatform of this invention.

FIG. 6A is a graph depicting the measurement of a wind Doppler speedprofile by the system and method shown in FIGS. 1-5 operating in adownward looking in-flight mode;

FIG. 68 is a graph depicting the measurement of the wind Doppler speedprofile by the system and method shown in FIGS. 1-5 operating in anupward looking ground based mode;

FIGS. 7A and 7B depict coordinate systems when a mobile airborneplatform aircraft roll pitch and yaw are zero (FIG. 7A) and when amobile airborne platform aircraft roil pitch are yaw are 10°, 0° and10°, respectively (FIG. 7B);

FIG. 8 depicts the radar beam pointing directions in the mobile airborneplatform coordinate system;

FIG. 9A depicts beam geometries in a ground based system with fourupward looking beams;

FIG. 9B shows the variation of radar beam directions during a 180° turnof a mobile airborne platform;

FIGS. 10A and 10B depict examples of the formation of cubic splines inaccordance with the AWiPPR system disclosed herein;

FIG. 11A shows the path of a mobile airborne platform conducting a 20°bank left turn;

FIG. 11B shows the vector wind velocity profile used in a simulation;

FIG. 11C shows simulation data for each of the eight beams generated bythe AWiPPR airborne wind profiling system and method disclosed herein;

FIG. 12A shows the results from the maximization of the evidence of thefunction of the smoothing scale and noise precision;

FIGS. 12B and 12C depicts slices across the evidence surface shown inFIG. 12A;

FIG. 13 depicts the least squares and maximum evidence of estimates ofvector wind velocity with a 20° bank turn test case shown in FIG. 11A;

FIG. 14 shows the effects of spline over fitting for the 20° hank turntest case;

FIGS. 15A-15B depict the Beckmann-Spizzichino back scatter model used inthe processing of the reflected radar signal in some examples of theinvention;

FIG. 16A depicts a theoretical back scatter in one region of the modelof FIG. 15 for a range of surface roughness parameters;

FIG. 16B is a graph depicting un-normalized back scatter levels overland and water observed by a test of the system disclosed herein;

FIGS. 17A and 17B are graphs depicting the effect of surface back scaleron the detection of clear air wind echoes;

FIG. 18 is a graph depicting the radar noise floor data observed in atest of the AWiPPR system and method disclosed herein;

FIG. 19 is a graph depicting transmittance weighted relative tobrightness temperature;

FIG. 20A is a graph depicting sky noise as a function of radarelevation;

FIG. 20B is a graph depicting the time variation of sky noise;

FIG. 21 is a chart depicting a mobile airborne platform flight leg in atest of the AWiPPR system disclosed herein;

FIG. 22 are plots of navigation data for a part of the flight leg shownin FIG. 21;

FIG. 23 are plots of navigation data from a second portion of the flightleg depicted in FIG. 21;

FIG. 24 is a pictorial representation of the aiming maneuver of theflight pattern shown in FIG. 21;

FIG. 25 is a graph of back scatter data observed during a test of theAWiPPR system disclosed herein;

FIG. 26 are plots showing data collected as the mobile airborne platformturns;

FIG. 27 are plots showing data collected and processed with Dopplermotion correction of the AWiPPR system and method disclosed herein;

FIG. 28 is an example showing observed contact Doppler velocities andaltitudes together with radar beam pointing directions;

FIG. 29 is a plot depicting vector wind velocity at different altitudesmeasured by the AWiPPR system and method disclosed herein;

FIG. 30 are graphs depicting vector wind velocity at different altitudesmeasured by the AWiPPR system and method disclosed herein compared toradiosonde data;

FIG. 31 are plots depicting a comparison of measured and projected beamDoppler velocity data;

FIG. 32 is a depiction of a radar echo observed from a cloud formation;

FIG. 33 are plots of radar echoes from 12 consecutive radar files duringa time period in which the cloud echoes of FIG. 32 were observed;

FIG. 34A is a graph showing vector wind velocity at different altitudesfrom balloon radiosonde data;

FIG. 34B is a graph showing equivalent potential temperatures atdifferent altitudes;

FIG. 34C is a graph depicting Doppler velocity relative to radar atdifferent altitudes above ground level;

FIG. 35 are plots depicting motion corrected radar measurements of cloudDoppler velocity compared to radiosonde measurements;

FIG. 36A is a plot depicting measured versus predicted ground echoDoppler velocity measurements compared to ground echo Doppler velocitymeasurements;

FIG. 36B is a similar plot of measured versus predicted with an aircorrection computed by least squares minimization;

FIG. 37A shows a mobile airborne platform flight path;

FIG. 37B depicts the velocity error surface for a 45° turn;

FIG. 37C depicts a system velocity error surface for a 90° turn;

FIG. 37D depicts cross-track, along-track, and up-down velocity errorsfor a range of a mobile airborne platform turn angles;

FIG. 38A is a graph depicting the effect of varying degrees ofuncertainty in wind field knowledge on projectile drift;

FIG. 38B is a graph depicting the effect of varying degrees ofuncertainly in wind field knowledge on projectile drop; and

FIG. 39 is a flow chart depicting one example of steps utilized by theAWiPPR system and method disclosed herein.

DETAILED DESCRIPTION OF THE INVENTION

Aside from the preferred embodiment or embodiments disclosed below, thisinvention is capable of other embodiments and of being practiced orbeing carried out in various ways. Thus, it is to be understood that theinvention is not limited in its application to the details ofconstruction and the arrangements of components set forth in thefollowing description or illustrated in the drawings. If only oneembodiment is described herein, the claims hereof are not to be limitedto that embodiment. Moreover, the claims hereof are not to be readrestrictively unless there is clear and convincing evidence manifestinga certain exclusion, restriction, or disclaimer.

There are several problems associated with conventional wind profilingsystems. First, mounting a ground based radar to a moving platform thatis operating at velocities much greater than the wind speed requires newdata processing as disclosed herein since the targets now smear invelocity space, and the velocity of the aircraft may often be greaterthan the unambiguous velocity of the radar.

Second, pointing the radar at the ground means that the background is nolonger at 30° K., but is rather closer to 300° K., which means that thebackground noise floor is much higher. This reduces the signal to noiseratio (SNR) of the targets even if the platform is not moving.

Third, when pointing in a downward direction, the radar now has a largetarget in the ground bounce that can swamp the dynamic range preventingsmall targets from being visible. Higher incidence angles now reduce therequired dynamic range of the system and allows the system to resolveboth the return from the ground and that from the small CAS targets.

Fourth, the radar has to account for the motion of the platform with 6degrees of freedom: 3 positions (x, y, and z) and orientation (roll,pitch, and yaw).

Fifth, the radar must find a way to aggregate the data from alldirections into a format and a method that allows the information to beinverted into a wind solution. This is non-trivial as the system is notstatic, data is not sampled regularly, data cannot be required to fit apredetermined format or pointing direction, data is not guaranteed atany sampling interval, and an inversion matrix can be ill-conceived.

One or more embodiment of the AWiPPR system 10 and the method thereof,disclosed herein is an innovative airborne wind profiling portable radarsystem and method which provides a solution to one or more of theproblems discussed above. AWiPPR system 10, FIG. 1-4, is mounted tomobile airborne platform 12, FIGS. 1-3, e.g., an aircraft or similartype airborne vehicle and determines vector wind velocity as a functionof altitude above ground level. As disclosed herein, vector windvelocity may be described by three quantities 1) scalar wind speed, 2)wind direction in the horizontal plane, and 3) vertical scalar windspeed.

AWiPPR system 10 includes one or more navigation units 14, FIGS. 1 and4, mounted to a mobile airborne platform 12. One or more navigationunits 14 are configured to produce navigation data including at leastthe position and orientation of mobile airborne platform 12. In oneexample, the one or more navigation units 14 are typically a globalpositioning system or similar type navigation system, as discussedbelow.

AWiPPR system 10 also includes a radar unit 16, FIGS. 1, 3 and 4,mounted and positioned to a mobile airborne platform 12, e.g., as shownin FIG. 1, or as shown in a protype of AWiPPR system 10, FIGS. 2 and 3,Radar unit 16 is configured to transmit wide-band frequency modulatedcontinuous wave radar signal 18, FIG. 1, in a downward direction frommobile airborne platform 12 towards ground 20 as shown and is configuredto continuously receive reflected signals 22 from each of a plurality ofclear air scatters CAS targets 24 or volumetric targets 26 and outputradar data, e.g., the output transceiver 28 by line 30 (discussedbelow). CAS is the result of any number of phenomena that can causereflection of a radar signal. The most prevalent of these is convectiveturbulence generated by solar heating of the ground, Other clear airscatters include mechanical turbulence and reflections from insects. Inaddition to the foregoing, radar unit 16 may detect volumetric scatterreflections from rain, snow, fog, and virga (rain that evaporates beforeit reaches the ground). In all cases, these sources produce localchanges in the index of refraction in the atmosphere. Ultimately, it isthe changes in index of refraction that the radar unit 16 detects.

Radar unit 16 is preferably a wide-band (WB) frequency modulatedcontinuous wave (FMCW) radar capable of detecting targets 24 CAS orvolumetric targets 26 in the convective boundary layer (CBL) of theatmosphere. See, e.g., U.S. Pat. No. 9,310,,481 incorporated herein bythis reference, for an example of radar unit 16 configured as a WBCASFMCW radar unit. Radar unit 16 preferably operates at a carrierfrequency, fc, at about 33.4 GHz (Ka band) with selectable pulse sweepwidths of 6 to 100 MHz. The size of the sweep width controls the rangeresolution of radar unit 16 and the maximum effective range of radarunit 16. Sweep width is preferably chosen to match the back-scatteringcharacteristics of the CAS targets 24 or volumetric targets 26. Radar 16is preferably configured to detects CAS targets 24 or volumetric targets26 up to and beyond the top of the convective boundary layer (CBL).Depending on time of the day and atmospheric stability, the top of theCBL is nominally about 1500 m but it can be as high as 2500 m. Radarunit 16 preferably detects CAS targets 24 or volumetric targets 26 whichmay include the turbulent motion of the air associated with theever-present hydrodynamic-thermodynamic instabilities in the atmosphere.These turbulent motions move with the mean vector wind velocity andreflections from these features can be used to determine the mean vectorwind velocity. Their prevalence is most pronounced during time periodswhen solar illumination is high and the atmospheric equivalent potentialtemperature profile has a negative gradient with respect to increasingaltitude, CAS targets 24 or volumetric targets 26 can be tracked overseveral radar altitude cells and they predominantly have an apparentupward component of motion that causes them to appear to accelerate awayfrom the radar. This apparent component of acceleration is caused by thefinite beam width of the radar beam. Radar unit 16 also may detect traceprecipitation, rainfall, snow, fog and clouds and other dull airphenomena.

Radar unit 16 is preferably adapted for use on mobile airborne platform12, FIGS. 1-3, so that the vector wind velocity profile could bemeasured from the top of the CBL down to ground 20, FIG. 1. Radar unit16 is preferably configured as downward-looking radar as shown. Airbornemeasurement of the CBL wind profile with radar unit 16 configured as adownward-looking radar is a much more challenging technical problem thanconventional ground based wind profile measurement using anupward-looking radar. Key technical challenges include the following:

-   -   1. Radar unit 16 has been successful in part because of its        large dynamic range and low noise floor. The noise floor for        system 10 on a temperature scale at near vertical elevation        angles is approximately 150° K. of which about 38° K. comes from        atmospheric sources. When the radar unit 16 is turned upside        down it is pointed at the ground with a nominal temperature of        300° K. This will cause the radar noise floor to increase from        150° K. to approximately 420° K. representing a net loss in        performance against weakly reflecting clear air targets of about        4.5 dB.    -   2. Backscatter echoes from ground 20 are 4-7 orders of magnitude        larger than CAS echoes and may cause saturation thereby limiting        the performance of radar unit 16.    -   3. In order to infer vector wind velocity from observed Doppler        echoes from mobile airborne platform 12 it is necessary to know        the vector velocity and orientation of the mobile airborne        platform 12 in a reference inertial coordinate system with a        high degree of accuracy and precision. This requires one or more        navigation units 14 be tightly coupled to the data collection        process of radar unit 16.    -   4. The character of the measured data of mobile airborne        platform 12 may be very different than data measured from a        conventional ground-based system. The algorithms for processing        airborne data need to be more sophisticated than conventional        ground based systems.

One or more embodiment of AWiPPR system 10 mounted on mobile airborneplatform 12 with radar unit 16 provides a solution to the abovetechnical challenges. A prototype of AWiPPR system 10 was tested. UsingAWiPPR system 10 and method thereof, the resulting data was processedand a vector wind velocity profile was determined that was found to bein reasonable agreement with vector wind velocity measured by a balloonlifted radiosonde nearby, as discussed in detail below.

In one design, radar unit 16, FIG. 3, is preferably a FMCW type radar orsimilar type radar unit. In operation, radar unit 16 is always on, andis continuously transmitting a wide-band frequency continuous wave radarsignal 18. FIG. 1. Radar unit 16 preferably includes a sweep generator32, FIG. 4, which provides the transmitted waveform, preferably a linearfrequency modulated (FM) sweep. Other waveforms may be used, such as aphase coded waveform, or a non-linear FM waveform. In one example, radarunit 16 may be configured as a frequency modulated continuous wave(FMCW) radar using a linear frequency sweep estimates the range andDoppler velocity of a target echo by using a form of fast-time slow-timeprocessing that produces a range-velocity matrix (RVM), AWiPPR system 10forms RVMs in the same way as the WBCAS FMCW radar system described inU.S. Pat. No. 9,310,481, incorporated by reference herein.

Radar unit 16 includes transceiver 28 which takes the baseband signaland converts it up to Ka band, filters the transmit signal, andamplifies transmit signal 18, FIG. 1. Transceiver 28, FIG. 4, receivesthe continuously reflected signal 22 from each of the plurality of CAStargets 24 or volumetric scatters targets 26, amplifies the receivedsignal, converts the signal to baseband, filters the baseband signal,and amplifies the baseband signal and outputs radar data by line 30 todata acquisition (DAQ) unit 34 (discussed below). The down-conversion ispreferably homodyne and may either be single side band or dual sideband.

Radar unit 16 also includes one or more antennas, e.g., antenna 20 aand/or antenna 20 b, FIGS. 1-4, e.g., a single antenna or dual antennas.In one design, separate linearly polarized antennas may be utilized toconnect to separate transmit and receive ports on transceiver 28. Inother designs, left and right-hand polarized antennas may be utilized, asingle antenna with a circulator, or a single antenna with separatepolarization feeds may be utilized. A single antenna solution withcirculator will require additional circuitry to cancel the antenna feedreflection and potentially any close-target returns.

AWiPPR system 10, FIGS. 1-4, also includes inertial management unit(IMU) 38, FIGS. 3 and 4. EAU 38, FIG. 4, is in communication with theone or more navigation units 14 and radar unit 16. IMU 18 is configuredto receive the navigation data output by the one or more navigationunits by line 40, FIG. 4, and determine the position and orientation ofradar unit 16 at a specific time and output IMU data by line 42. In oneexample, IMU 38 may include navigation unit 14, e.g., a globalpositioning service (GPS), and receives signals therefrom along withother inputs including roll rate sensors, inertial sensors,magnetometers, temperature and pressure sensors located in or on mobileairborne platform 12, IMU 38, or radar unit 16 to determine the positionand orientation of the radar unit 16 Careful measurements are taken totranslate the measurements of IMU 28 into antenna pointing directionsfor one or more antenna 20 a and/or 20 b, as discussed below.

AWiPPR system 10 also includes DAQ unit 34, FIG. 3, which is typicallypart of the electronics of radar unit 16. In one example, navigationunit 14 may be embedded in DAQ 34. DAQ 34 is in communication with radar16 and IMU 18 and is configured to receive and time align the radar dataoutput by transceiver 24 of radar unit 16 by line 30 and the IMU dataoutput by IMU 38 by line 42 for each reflected signal 22, FIG. 1, fromeach of the plurality of CAS targets 24 or volumetric targets 26 toprovide an antenna positioning direction for one or more antenna 20 aand/or 20 b, FIGS. 1-4, for each received reflected signal.

DAQ unit 34 is configured to process the time aligned radar data and IMUdata to determine a distance and Doppler velocity of each oldieplurality CAS targets 24 or volumetric targets 26 provide a range, avelocity, and an antenna pointing direction for each of the pluralityCAS targets 24 or volumetric scatters targets 26, and calculate a vectorwind velocity using the range, the velocity, and the antenna pointingdirection for each of the plurality of CAS targets 24 or volumetrictargets 26.

DAQ 34 is also configured to further correct the range, the velocity,and/or the antenna pointing direction of each of the plurality of CAStargets 24 or volumetric scatters targets 26 to accommodate for a motionshift in the data produced by one or more of a relative motion inorientation of mobile airborne platform 12, a Doppler spread in the RVM,and a ground echo.

DAQ 34 preferably digitizes the baseband signal and segments thewaveform to provide alignment to the transmit pulse. DAQ 34 records thedata to local memory 58 before processing the data. Field control system60 is a computer subsystem that provides a human machine interface tocontrol the functions of DAQ 34.

System 10 preferably includes a plurality of clocks, e.g., a clock inradar unit 16, a clock in inertial management unit 18, a clock in DAQ34, and a clock in one or more navigation units 14, as known by thoseskilled in the art. The clocks in system 10 are preferably synchronizedto each other and preferably have minimal phase noise. By locking theclocks together, system noise is confined to very specific regions whichcan then be removed. This allows for very long integration times andsignificant processing gain free from low level noise. By reducing phasenoise real dynamic range is increased.

In one example, a motion shift in the range, the velocity, and theantenna pointing direction does not include a Doppler wrap and the oneor more navigation units 14 is configured to generate a navigationcorrection for each reflected signal 22 by adding a speed of mobileairborne platform 12 provided by the one or more navigation units 14 tothe determine Doppler velocity for each of the plurality CAS targets 24or volumetric scatters targets 26 and rotating the range, the velocity,and the antenna pointing direction into a coordinate system centeredbeneath mobile airborne platform 12.

In one embodiment, DAQ 34 is responsive to the sparseness of theplurality of CAS targets 24, shifts in the position of the mobileairborne platform 12 over a predetermined measurement window, typicallycomprised of at least the amount of time from when wide-band frequencymodulated continuation wave radar signal 18 is transmitted and thereflected signals 22 from each of the plurality CAS targets 24 orvolumetric targets 26 is received by radar unit 16, and navigationcorrection is applied to a set of reflected signals from the pluralityof CAS targets 24 or volumetric targets 26 DAQ 34 is further configuredto infer a Doppler field vector for each of the plurality of CAS targets24 or volumetric scatters targets 26 as a set of three coupled cubicsplines derived from measured Doppler velocity data for the plurality ofCAS targets 24 or volumetric targets 26 using a non-parametric functionestimation, as discussed below.

The method of determining a vector wind velocity as a function ofaltitude above the ground on a mobile air platform of one embodiment ofthis invention includes providing navigation data including at leastpositioning and orientation of the mobile airborne platform, step 100,FIG. 5. A wide-hand frequency modulated continuous wave radar signal istransmitted in a downward direction from the mobile airborne platformtowards the ground, step 102. A reflected signal is received from eachof the plurality of CAS targets or volumetric targets and radar data isoutput, step 106. The position and orientation of a radar unit mountedand positioned on the mobile airborne platform is determined at aspecific point in time and position and orientation data is output, step108. The radar data and the position and orientation data for eachreflected signal from each of the plurality of CAS targets or volumetrictargets is time aligned to provide an antenna pointing direction foreach of the plurality of CAS targets or volumetric targets, step 110.The timed aligned navigation data and the position and orientation dataare processed to determine a distance and Doppler velocity for each ofthe plurality of CAS targets or volumetric targets and provide a range,a velocity, and antenna pointing direction for each plurality of CAStargets or volumetric scatters targets and calculate a vector windvelocity using the range, the velocity, and the antenna pointingdirection for each of the plurality of CAS targets or volumetrictargets, step 112. The range, the velocity, and the antenna pointingdirection of each of the plurality of CAS targets or volumetric targetsis further corrected to accommodate for a motion shift in data producedby one or more of a relative motion in orientation of the mobileairborne platform, a Doppler spread in the RVM and a ground echo, step114.

The result is AWiPPR system 10 and the method thereof efficiently andeffectively determines vector wind velocity as a function of altitudeabove ground level in mobile airborne platform 12. AWiPPR system 10 andthe method thereof provides for the use of airborne radar for detectingCAS targets or volumetric scatters targets or other features that createradar reflections. AWiPPR system 10 and the method there of usesaircraft navigation data to georeference CAS targets and the dotproducts of their velocity. AWiPPR system 10 and the method thereof usesaircraft navigation data to correct relative wind data. AWiPPR system 10and the method thereof provides a solution of inverseproblem/tomographic reconstruction to estimate vector wind velocityvector as a function of altitude. AWiPPR system 10 and the methodthereof operates in dull weather conditions, for example dust, fog,mist, virga, and snow. AWiPPR system 10 and the method thereof mayoperate in the Ka band (33.4 GHz) FMCW radar due to radar bandavailability and experimental results. AWiPPR system 10 and the methodthereof provides selectable pulse sweep widths of about 6 to about 200MHz to control range resolution and maximum range of radar and providesselectable pulse duration, small range bands, e.g., 3.125 meters. AWiPPRsystem 10 and the method thereof provides a very low-noise front end,e.g., less than about 2 dB Noise Figure. AWiPPR system 10 and the methodthereof also uses data selection by SNR thresholding and provides fordata selection and validation by outlier removal and data selection andvalidation by echo intensity and angle of incidence. AWiPPR system 10and the method thereof provides a solution of inverse problem usingmaximum a posteriori (MAP) cubic splines, parametrized by smoothingparameter and noise parameter and maximization of MAP probability overthe space of smoothing parameter and noise parameter, as discussed infurther detail below.

An example of the differences between RVMs formed by AWiPPR system 10during in-flight and post-flight testing is shown in FIGS. 6A and 6B.Both of the RVMs shown contain Doppler velocity echoes from the windprofile at the measurement location but the RVM recorded during flightoperations additionally contains a very strong echo from the grounddenoted by A in FIG. 6A as well as echoes from sidelobes in the radartransmit-receive beam pattern denoted by B and C. Although notindicated, the Doppler wind profile contains motion effects from mobileairborne platform 12 that have not been removed. An understanding of theprocessing step necessary to extract useful vector wind velocityinformation from an RVM measured on a moving platform requires anunderstanding of differences in the kinematics between moving and fixedplatform data collection.

FIG. 6A is a measurement of the wind Doppler speed profile with AWiPPRsystem 10 and the method thereof operating in a downward-lookingin-flight mode. FIG. 6B depicts an upward-looking ground-based mode. Themobile airborne platform 12 is operating at an altitude of approximately800 m. Winds above the aircraft cannot be measured with adownward-looking radar.

AWiPPR data processing of AWiPPR system 10 and the method thereof isbased upon the two interrelated coordinate systems shown in FIGS. 7A and7B. The first coordinate system of FIG. 7A is denoted by xyz. Thiscoordinate system is fixed with respect to the mobile airborne platform12. The x-axis runs from the tail of the mobile airborne platform 12 outthrough the nose of the mobile airborne platform 12 and the y-axis runsfrom the left wingtip of the mobile airborne platform 12 out through theright wingtip of the mobile airborne platform 12. The z-axis in the xyzcoordinate system is given by the curl of the x-axis unit vector withthe y-axis unit vector. Thus if the mobile airborne platform 12 is inlevel flight, the z-axis points straight up. The second coordinatesystem of FIG. 7B is denoted by XYZ. It is centered directly beneath themobile airborne platform 12. In the second coordinate system the X, Yand Z axes respectively point in the directions east, north and verticalThe mobile airborne platform 12 roll, pitch and yaw together with themobile airborne platform 12 vector velocity, v_(rador)={u_(x), u_(y),u_(z)}, are measured in the XYZ coordinate system. Since radar unit 16is affixed to the mobile airborne platform 12 the vector velocity of themobile airborne platform 12 is preferably denoted by v_(radar) ratherthan v_(aircraft). The height of radar unit 16 above the ground isdenoted by and is measured in the XYZ coordinate system. The followingroll pitch and yaw angle rotation conventions may be used in thediscussions below:

-   -   1. When roll is positive the right wing of the mobile airborne        platform 12 will tip towards the ground.    -   2. When pitch is positive the nose of the mobile airborne        platform 12 will tip towards the ground.    -   3. Yaw is the heading of the mobile airborne platform 12        measured with respect to true north. Positive yaw causes the        airplane to turn to the right

FIG. 7A and 7B shows AWiPPR system 10 and the method thereof coordinatesystems when (a) mobile airborne platform 12 roll, pitch and yaw arezero and (b) mobile airborne platform 12 roll, pitch and yaw arerespectively 10°, 0° and 10°. The height of mobile airborne platform 12above the ground, roll, pitch and yaw together with the radar velocityare measured in the XYZ coordinate system. The four lines shown in FIGS.7A and 7B represent four downward looking radar beams at azimuthallocations 0°, 90°, 180° and 270° measured with respect to the xyzcoordinate system.

One example of radar beam directions of radar unit 16 defined in the xyzcoordinate system are shown in FIG. 8. The specific direction in whichthe radar beam points in the xyz coordinate system is defined by theequation:

η _(beam)(θ,ϕ)={cos θsin ϕ, cos θcos ϕ,−sin θ}

In this equation the angle θ is measured positive down with respect tothe xy-plane and the angle ϕ is measured positive clockwise with respectto the y-axis, if the mobile airborne platform 12 undergoes roll, pitchand yaw respectively denoted by α_(r), β_(p) and γ_(y), then the radarbeam will point in the new direction using the equation:

η _(beam)(α, β_(p), γ_(y), θ, ϕ)=M _(yaw) M _(pitch) M _(roll) {cos θsinϕ, cos θcos ϕ, −sin θ}

where M_(roll), M_(pitch) and M_(yaw) are the axis rotation matricesdefined by

${M_{roll} = \begin{pmatrix}{\cos \mspace{11mu} \alpha_{r}} & 0 & {\sin \mspace{11mu} \alpha_{r}} \\0 & 1 & 0 \\{{- \sin}\mspace{11mu} \alpha_{r}} & 0 & {\cos \mspace{11mu} \alpha_{r}}\end{pmatrix}},{M_{pitch} = {\begin{pmatrix}1 & 0 & 0 \\0 & {\cos \mspace{11mu} \beta_{p}} & {\sin \mspace{11mu} \beta_{p}} \\0 & {{- \sin}\mspace{11mu} \beta_{p}} & {\cos \mspace{11mu} \beta_{p}}\end{pmatrix}\mspace{14mu} {and}}}$ $M_{yaw} = \begin{pmatrix}{\cos \mspace{11mu} \gamma_{y}} & {\sin \mspace{11mu} \gamma_{y}} & 0 \\{{- \sin}\mspace{11mu} \gamma_{y}} & {\cos \mspace{11mu} \gamma_{y}} & 0 \\0 & 0 & 1\end{pmatrix}$

Each of these rotations is performed with respect to the XYZ coordinatesystem. Roll is performed about the Y-axis. Pitch is performed about theX-axis and yaw is performed about the Z-axis. An example of the effectof roil and pitch on the orientation of the aircraft and the radar beamsis presented in FIG. 7B.

FIG. 8 shows the beam pointing directions in mobile airborne platform 12xyz-coordinate system. The angle ϕ is measured positive down withrespect to the xy-plane and the angle ϕ is measured positive clockwisewith respect to the y-axis.

The use of the WBCAS radar by radar unit 16 on mobile airborne platform12 has two primary effects on the performance of radar unit 16. First,motion of mobile airborne platform 12 produces Doppler spread in thevelocity signature of a target that is proportional to the speed of themobile airborne platform 12 times the width of the radar beam inradians. The amount of Doppler spread is independent of the direction ofplatform motion. Second, mobile airborne platform 12 motion shifts theobserved Doppler velocity of a target echo by an amount that depends onthe speed and heading of the moving mobile airborne platform 12.

There are two basic types of motions corrections that may be utilized byAWiPPR system 10 and method thereof. At low platform speeds whereDoppler wrap is not an issue, the correction for mobile airborneplatform 12 motion can be applied after an estimate of vector windvelocity has been made in a moving coordinate system relative to themobile airborne platform 11. In this case the proper post-processingcorrection for the motion of the mobile airborne platform 12 can befound by using the following procedure. Let (v_(x), v_(y), v_(z)) denotethe true values of wind speed in a fixed coordinate system and let(u_(x), u_(y), u_(z)) denote the wind speed values observed by radarunit 16 on a horizontally moving mobile airborne platform 12 with speedv_(radar) and at heading ψ_(rador) with respect to north. Observe thathorizontal motion does not affect the measurement of the verticalcomponent of wind speed. This implies that u_(z)=v_(z). The mobileairborne platform 12 speed, i v_(radar), s added to u_(y) in order tocorrect for forward motion in the mobile airborne platform 12 coordinatesystem, The velocity vector (u_(x), u_(y)+v_(radar), v_(z)) is rotatedin the platform coordinate system back into an EW-NS coordinate system.The result of this process is:

v _(x)=cos ψ_(radar) ·u _(x)+sin ψ_(radar) (u _(y) +v _(radar)), v_(y)=−sin ψ_(radar) ·u _(x)+cos ψ_(radar) (u _(y) +v _(radar)), v _(z)=u _(z).

When mobile airborne platform 12 speeds exceed about 50 mph, it will benecessary to use a correction procedure that directly addresses theproblem of Doppler wrap. The equation which relates the observed Dopplervelocity to the velocity of the wind and radar is:

V _(obs)=(V _(radar) −v _(wind))·η_(beam)

In principle, this equation can be rearranged into the form:

V _(obs) −v _(radar)·η_(beam) =−v _(wind)·η_(beam)

where all the quantities on the left-hand side of the equation are knownand the only unknown quantity on the right-hand side is the vector windvelocity. The difficulty arises from the fact that the speed of themobile airborne platform 12 will cause the observed Doppler velocityvalues to wrap due to the finite bandwidth effects of the radar digitalprocessor. Due to this problem, the observed values of Doppler velocityat the radar beam level must be motion corrected before a valid estimateof the wind speed can be obtained. The correct algorithm for doing thisis given by the following equation:

V _(obs) ^(corrected) =F _(wrap) [V _(obs) −F _(wrap)(v_(radar)·η_(beam))]=−v _(wind)·η_(beam)

In this equation V_(obs) is the Doppler velocity recorded by the radarsignal processor, v_(radar) is the vector velocity of the radar,v_(wind) is the vector velocity of the wind and is the pointingdirection of the radar beam. The function F_(wrap)(V) is defined by theequation:

F _(wrap)(V)=mod(V−V _(max),2V _(max))−V _(max)

where mod[V,V_(max)] denotes the modulo function on the interval (0.2V_(max)) and V_(max) is the maximum positive Doppler shift that can bemeasured by the radar's digital signal processor without producingDoppler wrap.

The action of the modulo function (mod) in the foregoing equation canbest be explained by way of an example. Suppose that radar unit 16 ismoving towards the north at a speed of 29 m/s and that the wind iscoming from the north at −10 m/s. Further suppose that radar unit 16 isusing a Fast Fourier Transform (FFT) signal processor that has anunambiguous Doppler velocity range of −11 m/s to 11 m/s. A northpointing radar beam with an elevation angle of 30 degrees will bephysically presented with a Doppler shift due to the combined effects ofthe wind and platform motion that is equal to (29+10)cos(30 deg) whichis 33.775 m/s. The actual observed Doppler shift at the output of theradar digital processor will be V_(obs)=mod[33.775−11, 22]−11=−10.255m/s. The contribution to wrapped Doppler shifting due to the motion ofthe radar unit 16 is given by mod[29 cos(30 deg)−11, 22]−11=3.114 m/s.If we apply this correction to the recorded Doppler velocity the resultis −13.34 m/s. But this converts to mod[−13.34−11, 22]−11=8.66 m/s. Ifradar unit 16 were not moving then the negative of the observed Dopplervelocity would be 10 cos(30 deg)=8.66 deg which agrees with thecorrection given by the foregoing equation.

Convenient ground based GWiPPR system typically uses four upward-lookingradar beams as a basis for the estimation of vector wind velocity frombeam Doppler velocity measurements. An example of an upward-lookingGWiPPR type measurement is shown in FIG. 6B, based on the beam geometryshown in FIG. 9A. The beams are elevated 80° up from the horizontal andpointed in the cardinal azimuthal directions north, east, south and west(cardinal directions are not required, but are convenient). Dopplervelocity data is recorded on each beam and inverted to find the vectorwind velocity as a function of altitude. The actual procedure used to dothis is disclosed in U.S. Pat. No. 9,310,481, incorporated herein bythis reference. A perfectly valid way of doing this inversion is to takethe Doppler velocity data measured on an individual beam and fit thisdata with a low order polynomial or a cubic spline. If this is doneusing a cubic spline with control points {z_(n), f_(n)}, n=1,2, . . . N,then the relationship between the measured Doppler data {z_(k),V_(k)},k=1,2, . . . , K and the unknown spline abscissas f={f₁, f₂, . . . ,f_(N)}^(T) is shown by equation:

V _(K×1) =B _(K×N) f _(N×1)

In this equation V={V₁, V₂, . . . , V_(K)}^(T) is the vector of measuredDoppler values and B is a matrix whose entries depend only on thealtitudes at which Doppler velocities were measured and the type ofspline used to represent the observed Doppler velocity profile, Theleast squares solution to this equation is f=(B^(T)B)⁻¹B^(T) V.

If this fitting process is carried out for each of the four radar beams,then the result will be a set of four Doppler profiles V^((i))(z),i=1,2,3,4. From these four profiles the vector velocity of the windv(z)={v_(x)(z),v_(y)(z),v_(z)(z)} can be computed from:

${v_{x}(z)} = {\frac{1}{2\mspace{11mu} \cos \mspace{11mu} \theta}\left\lbrack {{V^{(4)}(z)} - {V^{(2)}(z)}} \right\rbrack}$${v_{y}(z)} = {\frac{1}{2\mspace{11mu} \cos \mspace{11mu} \theta}\left\lbrack {{V^{(3)}(z)} - {V^{(1)}(z)}} \right\rbrack}$${v_{z}(z)} = {- {\frac{1}{4\mspace{11mu} \sin \mspace{11mu} \theta}\left\lbrack {{V^{(1)}(z)} + {V^{(2)}(z)} + {V^{(3)}(z)} + {V^{(4)}(z)}} \right\rbrack}}$

where θ is the elevation of each of the four radar beams above thehorizontal. An important point here is that the spline was fitted toobserved data. Both the measured data on a beam V_(k) and the splineabscissas for that beam f_(n) are measured in units of Doppler velocity.Additionally, the solution was enabled by the symmetry of the four beamsystem and the ability if necessary to monitor these beams for anextended time period in order to acquire an adequate amount of data.This approach does not extend to the AWiPPR problem because the 6 degreeof freedom motion of the mobile airborne platform 12 causes a continuousvariation in beam pointing directions. An example of this is shown inFIG. 9B. FIG. 9B depicts an idealized 180 deg turn of an aircraft flyingat constant altitude. As mobile airborne platform 12 approaches theturn, its roll angle decreases from 0° to −30°. This drops the left wingin preparation for the circular part of the turn. During the actualturn, the mobile airborne platform 12 maintains a constant roll angle of−30° while its yaw angle smoothly changes from 0° to −180°. Upon exitingthe turn, the roll angle increases from −30° to 0° with a constant yawangle of −180°. At all times the pitch angle is 0°. The radar beamrelative to the mobile airborne platform 12 is 0° of rotation and 65° ofdepression. The black downward pointing arrows in FIG. 9B indicate theradar beam directions at six positions along the turn. The length ofthese arrows is the slant range from the mobile airborne platform 12 tothe ground 20.

The approach for AWiPPR system 10 and method thereof disclosed hereinfor addressing the complexity of the airborne measurement relative toground measurement is based upon representing the wind vector field as aset of three coupled cubic splines and inferring (not fitting) thesecoefficients from the observed Doppler data over many beams and at manyaltitudes. Conceptually, AWiPPR system 10 and method thereof performsthe following:

v _(x)(z)⇒(z _(n) ,f _(n) ^((x))),

v _(y)(z)⇒(z _(n) ,f _(n) ^((y))),

v _(z)(z)⇒(z _(n) f _(n) ^((z))),

n=1,2, . . . , N

Wherein the symbol “⇒” means ‘represented by’. These three splines sharethe same control point abscissas but have different control pointordinates. The process that AWiPPR system 10 and the method thereof usesto infer the control point ordinates from the Doppler velocitymeasurements is a form of nonparametric function estimation. It is muchmore complicated than “fitting” a spline to data, AWiPPR system 10 andthe method thereof builds upon the concept of representing a scalarfunction by a natural cubic spline, as discussed below.

The construction of a cubic spline representation for a function f(z)that represents the wind Doppler velocity profile observed on a radarbeam begins with representing the second derivative of f(z) as apiecewise continuous linear function, In FIG. 10A as shown below, thevalues of the second derivative of f(z) at the abscissa points z_(j),j=1,2, . . . ,N are denoted by m_(j) with h_(j+1) defined to beh_(j+1)=z_(j+1)−z_(j). The second derivative of this function on theinterval z_(j)≤z≤z_(j+1) is shown by the equation:

${S^{''}(z)} = {{m_{j} + {\frac{m_{j + 1} - m_{j}}{h_{j + 1}}\left( {z - z_{j}} \right)}} = {{\frac{z_{j + 1} - z}{h_{j + 1}}m_{j}} + {\frac{z - z_{j}}{h_{j + 1}}m_{j + 1}}}}$

In the above equation, S(z) is used as an alternate name for the cubicspline f(z). Integrating S″(z) twice yields a cubic polynomial that hascurvature on the interval (z_(j),z_(N)) that is less than any othertwice-differential function on the interval. The results of these twointegrations are shown in the FIG. 10B. The points (z_(j),f_(j)) arereferred to as the control points (also called pivot points) of thespline. Application of the requirement that S(z) must pass through thepoints (z_(j),f_(j)) and (z_(j+1),f_(j+1)) for j=2,3, . . . , N−1 leadsto the representation of the cubic spline S(z) shown in the equationbelow:

${S_{j}(z)} = {{\frac{m_{j + 1}h_{j + 1}^{2}}{6}\left\lbrack {\left( \frac{z - z_{j}}{h_{j + 1}} \right)^{3} - \left( \frac{z - z_{j}}{h_{j + 1}} \right)} \right\rbrack} + {\frac{m_{j}h_{j + 1}^{2}}{6}\left\lbrack {\left( \frac{z_{j + 1} - z}{h_{j + 1}} \right)^{3} + \left( \frac{z - z_{j}}{h_{j + 1}} \right) - 1} \right\rbrack} + {f_{j + 1}\left( \frac{z - z_{j}}{h_{j + 1}} \right)} + {f_{j}\left( \frac{z_{j + 1} - z}{h_{j + 1}} \right)}}$

At this point the spline pivot abscissas, z_(j), and their spacings,h_(j), are assumed to be known. The values of the pivot point ordinatesf_(j) and their second derivatives, m_(j), remain to be related to oneanother. Continuity of the first derivative of S(z) at the points(z_(j),f_(j)) for j=2,3, . . . , N−1 supplies N−2 equations. Theadditional requirement, that the second derivative of S(z) is zero atz_(l) and z_(N), supplies an additional two equations. This choiceproduces a form of spline that is referred to as a natural spline.Natural splines are terminated at inflection points.

FIGS. 10A and 10B show the formation of a cubic spline. FIG. 10A showsthe second derivative of the function f(z), and FIG. 10B shows thefunction f(z).

At this point the following information about the cubic spline f(z):

m₁ = 0${{{\frac{h_{j}}{6}m_{j + 1}} + {\frac{h_{j} + h_{j + 1}}{3}m_{j}} + {\frac{h_{j + 1}}{6}m_{j + 1}}} = {\frac{f_{j + 1} - f_{j}}{h_{j + 1}} - \frac{f_{j} - f_{j + 1}}{h_{j}}}},{j = 2},3,\ldots \mspace{14mu},{N - 1}$m_(N) = 0

This system of equations can be written in matrix format:

M _(N×N) m _(N×1) =F _(N×N) f _(N×1)

where the vector m={m₁, m₂, . . . , m_(N)}^(T) and the vector f of pivotpoint ordinates is f={f₁, f₂, . . . , f_(N)}^(T). The sampling matricesM and F are defined via:

$M_{N \times N} = {\frac{1}{6}\begin{bmatrix}{2h_{2}} & 0 & 0 & 0 & \ldots & \ldots & 0 \\h_{2} & {2\left( {h_{2} + h_{3}} \right)} & h_{3} & 0 & 0 & \; & \; \\\; & h_{3} & {2\left( {h_{3} + h_{4}} \right)} & h_{4} & 0 & \ldots & 0 \\0 & \ldots & \ldots & 0 & h_{N - 1} & {2\left( {h_{N - 1} + h_{N}} \right)} & h_{N} \\0 & 0 & 0 & 0 & \ldots & 0 & {2h_{N}}\end{bmatrix}}$   and $F_{N \times N} = \begin{bmatrix}0 & 0 & 0 & 0 & \ldots & \ldots & 0 \\\frac{1}{h_{2}} & {- \frac{h_{2} + h_{3}}{h_{2}h_{3}}} & \frac{1}{h_{3}} & 0 & 0 & \; & \; \\0 & \frac{1}{h_{3}} & {- \frac{h_{3} + h_{4}}{h_{3}h_{4}}} & \frac{1}{h_{4}} & 0 & \ldots & 0 \\0 & \ldots & \ldots & 0 & \frac{1}{h_{N - 1}} & {- \frac{h_{N - 1} + h_{N}}{h_{N - 1}h_{N}}} & \frac{1}{h_{N}} \\0 & 0 & 0 & 0 & \ldots & 0 & 0\end{bmatrix}$

The matrix M_(N×N) is nonsingular and has an inverse. The existence ofthis inverse allows us to write the vector m of second derivative valuesof the spline f(z) in the form of the equation:

m _(N×1) =M _(N×N) ⁻¹ F _(N×N) f _(N×1)

The remaining step in the application of a spline to a practical problemis to determine the unknown spline ordinate points f={f₁, f₂, . . . ,f_(N)}^(T) from a set of altitude-velocity data points (ξ_(i), V_(i))for i=1,2, . . . , N_(d) where N_(d) denotes the number of data points.To do this a set of spline abscissa values {z₁, z₂, . . . , z_(N)} ischosen. There are three requirements constraining this choice. First ofall the minimum spline abscissa value, z₁, must be equal to the minimumof the data, ξ_(i), values. Second the maximum spline abscissa valuemust be equal to the maximum of the data, ξ_(i), values. Finally, eachof the abscissa intervals (z_(j)z_(j+1)) must be populated by at leastone of the data abscissa values ξ_(i).

If the observed data abscissa ξ_(i) lies in the spline abscissa interval(z_(j),z_(j+1)) and if the spline f(z) defined by the pivot points(z_(i), f_(i)) represents a good fit to the observed data then it shouldbe approximately true that:

$V_{i} = {{\frac{m_{j + 1}h_{j + 1}^{2}}{6}\left\lbrack {\left( \frac{\xi_{i} - z_{j}}{h_{j + 1}} \right)^{3} - \left( \frac{\xi_{i} - z_{j}}{h_{j + 1}} \right)} \right\rbrack} + {\frac{m_{j}h_{j + 1}^{2}}{6}\left\lbrack {\left( \frac{z_{j + 1} - \xi_{i}}{h_{j + 1}} \right)^{3} + \left( \frac{\xi_{i} - z_{j}}{h_{j + 1}} \right) - 1} \right\rbrack} + {f_{j + 1}\left( \frac{\xi_{i} - z_{j}}{h_{j + 1}} \right)} + {f_{j}\left( \frac{z_{j + 1} - \xi_{i}}{h_{j + 1}} \right)}}$

But since it is the case that m=M⁻¹Ff, it follows that the measuredvalue V_(i) can be written in the form V_(i)=b_(i1)f₁+b_(i2)f₂+ . . .+b_(iN)f_(N) where the coefficients b_(ij) depend only on the ξ_(i) andz_(j). This leads at once to the following matrix equation:

V _(N) _(d) _(×1) =B _(N) _(d) _(×N) f _(N×1)

The foregoing equation represents the forward relationship betweenobserved scalar velocity data and the natural cubic spline thatrepresents the data. If the number of data points is such that N_(d)>Nthen the system of equations is over determined. If the square matrixB^(T)B is not ill-conditioned then the least squares solution to the setof over-determined equations is given by the equation:

f _(N×1)=(B _(N) _(d) _(×N) ^(T) B _(N) _(d) _(×N))⁻¹ B _(N) _(d) _(×N)^(T) V _(N) _(d) _(×1)

The relationship between the observed Doppler velocity of a clear airscatter echo, the velocity of the radar and the velocity of the wind isfound using the equation:

U _(i)=(v _(rador) −v _(wind))η _(beam)(α_(i), β_(i), γ_(i), θ_(i),ϕ_(i))=(v _(radar) −v _(wind))η _(i)

where i is an index that runs across all the observed data. Thisequation results from computing the negative of the time rate of changeof the length of the radar ray path from radar unit 16, FIGS. 1 and 4,to the CAS targets 24, FIG. 1, or volumetric targets 26, and back toradar unit 16. If the data has been motion compensated this equationbecomes:

V _(i)=−v_(wind) η _(i)={η_(i) ^((x)) v _(x),η_(i) ^((y)),η_(i) ^((z)) v_(z)}

where η _(i)={η_(i) ^((x)), η_(i) ^((y)), η_(i) ^((z))}. The threescalar components of the vector wind velocity {v_(x),v_(y),v_(z)} can berespectively represented by the splines (z_(j),f_(j) ^((x)),(z_(j),f_(j) ^((y))) and z_(j),f_(j) ^((z))) where j=1,2, . . . , N. Inlight of the information matrix B defined previously,

V _(i)=−η_(i) ^((x))(b _(il)f_(l) ^((x)) + . . . b _(iN) f _(N)^((x)))−η_(i) ^((y))(b _(il) f _(l) ^((y)) + . . . b _(iN) f _(N)^((y)))−η_(i) ^((z))(b _(il) f _(l) ^((z)) + . . . b _(iN) f _(N)^((z))).

The preceding equation can be written in the vector-matrix form:

V _(N) _(d) _(×1) =A _(N) _(d) _(×3N) f _(3N)

where f={f₁ ^((x)), . . . , f_(N) ^((x)), f₁ ^((y)), . . . , f_(N)^((y)), f₁ ^((z)), . . . , f_(N) ^((z))}^(T) is the vector of unknownspline coefficients and V={V₁, . . . , V_(N) _(d) }^(T) is the vector ofmeasured Doppler velocity values,

If the square matrix A^(T)A is not ill conditioned, then the unknownspline coefficients in a least squares sense are given byf_(1S)=(A^(T)A)⁻¹A^(T)V.

To avoid this potential difficulty, the spline functions f^((x))(z),f^((y))(z) and f^((z))(z) have either smooth first or secondderivatives. This assumption implies that the prior probability densityof the unknown spline coefficients f can be written in the form

${p\left( f \middle| \mu \right)} = {\frac{1}{Z(\mu)}{\exp\left( {{- \frac{\mu}{2}}f_{3N \times 1}^{T}P_{3N \times 3N}f_{3N \times 1}} \right)}}$

where Z(μ) is a normal constant and μ is a hyper-parameter that can beestimated from the radar data and the IMU data disclosed above For thespecial case of N=4 equally spaced pivot points per spline and a firstderivative smoothness prior P_(3N×3N) takes the form:

$P_{12 \times 12} = {\frac{1}{h}\begin{bmatrix}{2 + ɛ} & {- 2} & 0 & 0 & 0 & \ldots & 0 \\{- 2} & {4 + ɛ} & 2 & 0 & 0 & \ldots & 0 \\0 & {- 2} & {4 + ɛ} & {- 2} & 0 & \ldots & 0 \\0 & \ldots & 0 & 0 & {- 2} & {4 + ɛ} & {- 2} \\0 & \ldots & \ldots & \ldots & 0 & {- 2} & {2 + ɛ}\end{bmatrix}}$

where h is the constant spacing between the pivot point abscissas and εis a small positive number that will eventually scale out in theanalysis that follows. The normalization constant Z(μ) is defined by:

${Z^{- 1}(\mu)} = {\left( \frac{\mu}{2\; \pi} \right)^{3{N/2}}\left( {ɛ{\prod\limits_{n = 1}^{{3N} - 1}\lambda_{n}}} \right)^{1/2}}$

where λ₁, λ₂, . . . , λ^(3N−1) are the nonzero eigenvalues of the firstderivative smoothing matrix P with ε set to zero.

The likelihood of obtaining the data set V={V₁, V₂, . . . , V_(N) _(d) }is equal to the product of the likelihood of the individual datasamples, From this it follows that:

${P\left( {{D\sigma},f} \right)} = {\prod\limits_{i = 1}^{N_{d}}{\frac{1}{\sqrt{2\; \pi \; \sigma^{2}}}{\exp \left\lbrack {{- \frac{1}{2\; \sigma^{2}}}\left( {V_{i} - {\sum\limits_{n = 1}^{3N}{a_{m}f_{n}}}} \right)^{2}} \right\rbrack}}}$

where the symbol D denotes the data set V={V₁, V₂, . . . , V_(N) _(d) }and σ is the noise associated with the measurement of Doppler velocity.In the analysis that follows it will be convenient to make the change ofvariable β=1/σ² where β is the precision of the noise. Large precisionequates to small noise and vice versa. Assuming that the noise parameterβ is unknown and that both μ and β need to be determined from the dataD. The shorthand vector-matrix form of P(D|β,f) is:

${P\left( {{D\beta},f} \right)} = {\frac{1}{Z_{D}(\beta)}{\exp \left\lbrack {{- \frac{\beta}{2}}\left( {V - {Af}} \right)^{T}\left( {V - {Af}} \right)} \right\rbrack}}$

where Z_(D)(β) is the normalization constant Z_(D)(β)=(2π/β)^(N) ^(d) ².

Bayes theorem enables the posterior distribution of the unknown pivotpoints f is be written in the form:

${P\left( {{fD},\mu,\beta} \right)} = {\frac{{likelihood} \times {prior}}{evidence} = \frac{{P\left( {{Df},\beta} \right)}{P\left( {f\mu} \right)}}{P\left( {{D\mu},\beta} \right)}}$

where P(D)|μ,β) is referred to as the evidence. The posteriorprobability distribution for the spline coefficients can be written inthe form:

${P\left( {{fD},\mu,\beta} \right)} = \frac{\exp \left\lbrack {{- \frac{1}{2}}{\Phi_{M}\left( {f,\mu,\beta} \right)}} \right\rbrack}{Z\left( {\mu,\beta} \right)}$where Φ_(M)(f, μ, β) = μ f^(T)Pf + β(V − Af)^(T)(V − Af)

is a model representing the combined effects of smoothing anddata-spline representation mismatch. The normalization coefficientZ(μ,β) is:

Z(μ,β)=∫_(−∞) ^(∞)exp[−½Φ_(M)(f,μβ)]d ^(3N) f

Since Φ_(M)(f,μ,β) is quadratic in the spline coefficient vector f itfollows that:

Φ_(M)(f,μ,β)=μf ^(T) Pf+β(f−f _(ML))^(T) Q(f−f _(ML))+βR _(ML)

where Q=A^(T)A is the 3N×3N spline precision matrix and f_(ML) is theset of spline coefficients that minimize the squared summed differencesbetween the data V and the model Af. Since minimizing the differencebetween the data and the model is equivalent to maximizing thelikelihood, the spline coefficients f_(ML) are referred to as themaximum likelihood estimate of f. The quantity R_(ML) denotes themaximum likelihood residual sum of squares and is defined by theequation:

R _(ML)=(V−Af _(ML))^(T)(V−Af _(ML))

An alternate relation of the smoothing data-spline representation modelΦ_(M)(f,μ,β) is:

Φ_(M)(f,μ,β)=(f−f _(MP))^(T) H(f−f _(MP))+R _(MP)

where f_(MP) denotes the most probable set of spline coefficients, H isthe precision matrix of the posterior distribution of the splinecoefficients f, and R_(MP) is a residual term that is the posterioranalogy to R_(ML). Equating terms in the second power of f between thetwo representations leads to:

H=μP+βQ

Equation terms in the first power of f gives:

Hf _(MP) =βA ^(T) Af _(ML) =βA ^(T) V, f _(MP) =βH ⁻¹ A ^(T) V

Evaluation of the two representations of Φ_(M)(f,μ,β) at f=f_(MP)produces:

R _(MP)(μ,β)=Φ_(M)(f _(MP),μ,β)=μf _(MP) ^(T) Pf _(MP) β|V−Af _(MP)|²

With these normalizations in hand the normalization function Z(μ,β) cannow be written as the Gaussian integral:

Z(μ,β)=exp[−½Φ_(M)(f _(MP),μ,β)]∫⁻²⁸ ^(∞)exp[−½(f−f _(MP))^(T) H(f−f_(MP))]d ^(3N) f

By way of an analogy to a multidimensional Gaussian distribution we cannow write:

Z(μ,β)=exp[−½Φ_(M)(f _(MP),μ,β)]|det(H)|^(−1/2)

A simple rearrangement of terms in the Bayes' formula leads to thefollowing representation for the evidence:

${P\left( {{D\mu},\beta} \right)} = {\frac{{P\left( {{Df},\beta} \right)}{P\left( {f\mu} \right)}}{P\left( {{fD},\mu,\beta} \right)} = \frac{Z\left( {\mu,\beta} \right)}{{Z(\mu)}{Z_{D}(\beta)}}}$

If we substitute our definitions for the three normalization functionsZ(μ), Z_(D)(β) and Z(μ,β) into the above formula and discard constantsthat do not depend on μ or β then we obtain the following representationfor the evidence P(D|μ,β)

${P\left( {{D\mu},\beta} \right)} = \frac{\left( {\mu^{3{N/2}}{\exp \left\lbrack {{- \frac{1}{2}}\mu \; f_{MP}^{T}{Pf}_{MP}} \right\rbrack}} \right)\left( {\beta^{N_{d}/2}{\exp \left\lbrack {{- \frac{1}{2}}\beta {{V - {Af}_{MP}}}^{2}} \right\rbrack}} \right)}{{\det \left( {{\mu \; P} + {\beta \; A^{T}A}} \right)}^{1/2}}$

where the most probable set of pivot points is defined by:

f _(MP)=β(μP+βA ^(T) A)⁻¹ A ^(T) V

The left hand factor in the numerator of our representation for theevidence is of the form μ^(3N/2)exp(−C₀μ) where C₀ is a positiveconstant that depends on the spacing of the pivot points but not upontheir ordinate values. As μ→0 or as μ→∞ the left-hand factor goes tozero and has maximal value at the solution to the equationμ^(3N/2-1)exp(−C₀μ)=0. The right-hand factor in the numerator of ourevidence representation is of the form β^(N) ^(d) ^(D)exp(−C₁β) where C₁is a positive constant that depends upon the mismatch between the data Vand the forward model Af_(MP) as characterized by the distance|V−Af_(MP)|². As β→0 or as β→∞ the right-hand factor goes to zero andhas maximal value at the solution to the equation β^(N) ^(d)²⁻¹exp(−C₁β)=0. Coupling between smoothness μ and noise precision β iscontrolled by the factor det(μP+βA^(T)A)^(1/2).

The preceding two equations can be numerically solved to find the valuesof the smoothing parameter p and the noise parameter β that maximize theevidence P(D|μ,β). If we denote these two values by μ_(E) and β_(E) thenour final estimate of the spline ordinates is:

f _(E) =βE(μ_(E) P+β _(E) A ^(T) A)⁻¹ A ^(T) V

If det(β_(E)A^(T)A) is large in comparison to det(μ_(E)P) then theresulting estimate for the pivot points of the spline is just the leastsquares solution,

The following wind example is based upon simulated AWiPPR data, Anexample using actual airborne data collected with the AWiPPR system 10and the method thereof is presented later, The scenario considered hereis vector wind velocity measurement with an AWiPPR system collectedduring a time period when the aircraft is conducting a 20 deg bankedleft turn as shown in FIG. 11A. The points labeled 1-8 indicate theaircraft position at 8 equally spaced positions on the turning path thatis indicated by the circle. The dashed lines indicate mobile airborneplatform 12 position with respect to a fixed ground based coordinatesystem in which vector wind velocity is measured. The mobile airborneplatform 12 is using a single-beam radar system with beam steeringdirection specified by ϕ=135 deg and θ=80 deg as described above. Theradar-beam directions are indicated by the arrows. The vector windvelocity profile used in the simulation is shown in FIG. 11B. The windprofiles extend up from the ground to a maximum elevation of 1600 m. Thesimulated data for each of the eight beams is shown in FIG. 11C. Thisdata has been generated by assuming that beam Doppler velocitymeasurements have a normally distributed random error with a standarddeviation of σ=0.85 m/s, The curves shown in FIG. 10C are not fits tothe beam data. They have been computed by finding the least squaressolution f_(LS)=(A^(T)A)⁻¹A^(T)V and projecting this solution onto eachof the 8 beam directions shown in FIG. 10A. In computing the leastsquares solution the number of pivot points for each velocity componenthas been chosen to be N=7 for a total 3N=21 pivot points.

FIGS. 12A and 12B shows evidence maximization for the 20 degree bankedturn test case. FIG. 12A shows the results from the maximization of theevidence as a function of the smoothing scale μ and noise precision β.

The evidence is maximized by using μ_(E)=211.35 and β_(E)=1.26. Thisvalue of β corresponds to σ_(E)=0.89 m/s which is very close to the 0.85values used to generate the simulated data. Slices across the evidencesurface shown in FIG. 12A are shown in FIGS. 12B and 12C. The ordinatefor each of these plots is log evidence. FIG. 13 compares the groundtruth vector wind velocity to the estimated values produced by thenonparametric function estimation process. The curves are the groundtruth vector wind velocity profiles from FIG. 11B. The least squaressolutions in the x, y and z directions are indicated by the larger dots.The smooth spline solution at intermediate points has been plotted ontop of the larger dots as a solid line of corresponding color. Thesmaller dots represent the solution from the maximum evidence procedure.The smaller dots are connected by dashed lines representing the splinesolution at intermediate points between the maximum evidence pivotpoints. A close look reveals the dashed lines resulting from theevidence maximization procedure are slightly better centered over theground truth plots. FIG. 14 shows the effect of changing from N=7 toN=20 pivot points. Now the value of the evidence maximization procedureshows itself The least squares solution is highly oscillatory as aresult of overrating the data. Evidence maximization corrects thisproblem.

FIG. 13 shows least squares and maximum evidence estimates of vectorwind velocity for the 20 degree banked turn test case, FIG. 14 show theeffects of spline overfitting for the 20 degree banked turn test case.

The electronics of AWiPPR system 10 are designed to be sensitive enoughto see the temperature of the cold sky and they have the capacity tomeasure the angular variation of sky temperature caused by the decreaseof radar absorption with increasing altitude. When radar unit 16 isupside down it sees the warm earth and additionally receivesbackscattered energy from the ground. It is this latter factor that isnow discussed here.

Backscattered energy from the ground is governed by a clutter form ofthe familiar radar equation and is proportional to the dimensionlessbackscatter coefficient σ₀(θ) where θ is the angle of incidence of theradar beam with respect to the vertical, The quantity σ₀(θ) representsthe fraction of incident power of a scattering surface element of areathat is scattered back to the radar receiver and as such it is adimensionless quantity.

Beckmann and Spizzichino (1987) developed a scattering model that hasproven to be effective at describing radar back scattering for variousfrequency-bands including the Ka band that the AWiPPR radar operates in.Various authors including Blake (1991), Campbell (2002), Barton (2013)and Rees (2013) have expanded and augmented the Beckrnann-Spizzichinomodel but the basic form has remained essentially the same. The essenceof the model is that dependence of the backscatter coefficient σ₀(θ) onθ can be divided into three angular regions. The first angular regionbegins at normal incidence (θ=0) and extends out a few tens of degrees.It is a region of quasi-specular scattering. Backscatter in this regioncan be quite high and can strongly couple to the radar via sidelobes.Smoother surfaces produce high peak backscatter levels at normalincidence but with angular decay with increasing θ that is much morerapid. Beyond this first region is a plateau region in which backscattervaries more slowly with changes in the angle of incidence. Backscatterin the plateau region tends to vary like σ₀(θ)=μ cos θ or σ₀(θ)=μ cos²θwhere μ is a constant dependent on the surface type. In this lattercase, the scattering is said to be Lambertian. Beyond the plateau regionthere is a third region near grazing incidence (θ≅89 deg) in whichbackscatter rapidly decreases. This region is referred to as theinterference region because of cancellation effects between scatteringpaths that differ by one surface bounce. These three regions aredepicted in FIG. 15A. The third region is of little interest to downwardlooking airborne radar and will not be discussed further.

In the first angular region extending from normal incidence out to a fewtens of degrees, Beckmann and Spizzichino (1987) make three assumptionsthat lead to an analytic solution for σ₀(θ). First they assume that thescattering surface is rough with heights that are normally distributedwith standard deviation h₀. Second they assume that the normalizedautocorrelation surface of the rough surface is given byC₀(x,Δx)=exp(−x²/≢x²) where x is a separation distance and Δx iscorrelation length.

Finally, they assume that the standard deviation of the surface heightirregularities is large in comparison with the space wavelength λ/cos θwhere λ is the wavelength of the radar carrier frequency. These threeassumptions lead to the following form for backscatter in region 1:

${\sigma_{0}(\theta)} = {\cot^{2}\beta_{0}{\exp \left( {- \frac{\tan^{2}\theta}{\tan^{2}\beta_{0}}} \right)}}$

where tan β₀=2h₀/Δx. The quantity tan β₀ can be interpreted as the ratioof the vertical scale of roughness to the horizontal scale of roughness.Campbell (2002) and Rees (2013) make slightly different statisticalassumptions about the rough surface but arrive at almost identicaltheoretical results.

Computations with this theoretical backscatter model are shown in FIG.16A for a range of values of the roughness parameter β₀. When β₀ issmall the peak scattering is clearly higher and confined to a smallerangular region. FIG. 16B shows observed unnormalized backscatter levelsover land and water observed by the AWiPPR system, The upper pointsindicate measurements made over land at an altitude of 740 m. The middleand lower points indicate measurements made over water at 740 m and 1600m respectively. The curved lines in are of the same form as thetheoretical scattering model defined by the foregoing equation. The overland data in FIG. 16B have been fitted with a straight line for visualemphasis. The distance between the curves in is approximately 10 dB.Surface backscatter theoretically falls off as 30log(R) where R is rangeIn this case 30log(1600/740)=10 dB is in exact agreement withtheoretical value.

An effective way to estimate the performance of a radar is to computethe matched filter signal to noise ratio SNR=t_(P)P_(scat)/N₀ wheret_(P) the coherent pulse length of the radar, P_(scat) is the receivedsignal power of the echo that is scattered back to the receiver,N₀=T_(system)k_(D) is the receiver noise power spectral density,T_(system) is the radar system noise temperature (not physicaltemperature) and k_(B)=138×10⁻²³ J/deg K is Boltzmann's constant. For anFMCW radar the coherent pulse length is t_(P)=N_(stack)T_(m) whereN_(stack) is the number of pulses, each of length T_(m) that are used toestimate the Doppler velocity of a target.

The backscatter power from the ground p_(scan)(θ) as a function of angleof the incidence θ at ground level can be computed (Blake, 1991) via thefollowing:

${{P_{scan}(\theta)} = {P_{t}\frac{\lambda^{2}G^{2}}{\left( {4\; \pi} \right)^{3}R^{4}}{A_{scat}(\theta)}{\sigma_{0}(\theta)}}},\mspace{14mu} {{A_{scat}(\theta)} = {\varphi_{beam}{R\left( \frac{c}{2B} \right)}\sec \; \theta}}$

where P_(i) radar is the transmit power, λ is the wavelength of theradar carrier frequency, G is antenna gain, R=z_(P)secθ is radar slantrange to the ground for a radar located at altitude z_(r) above thesurface of the earth, A_(scat) is the effective scattering area,ϕ_(beam) is the horizontal beam width of the system, c is the speed oflight, B is the bandwidth of the transmit pulse and σ₀(θ) is backscatteras a function of angle of incidence. For the AWiPPR system appropriatevalues are N_(stack)=256, T_(m)=190×10⁻⁶ sec, P_(i)=3.5 W, λ=9 mm,10logG=37 dB and B=48 MHz. The specific backscatter model used use inthe computations is shown in FIG. 15A.

The effect of backscatter from the ground on AWiPPR performance isillustrated in FIG. 17A. The upper curve in is the estimated peakbackscatter echo energy relative to k_(B) as a function of angle of theincidence θ at ground level for an AWiPPR system operating at a heightof z_(r)=800 m above the ground. Normalizing echo energy by Boltzmann'sconstant k_(B) is very convenient since it allows direct comparison ofthe echo energy to thermal noise energy measured on a temperature scale.Echoes with relative energy

t _(P) P _(scan)(θ)/k _(B) <T _(system)

are not detectable due to thermal noise masking. The AWiPPR radar'ssystem noise is in the 200-300° K. range.

The lower curve is relative echo energy diminished by a factor of 10⁻⁷corresponding to the estimated 70 dB dynamic range of the AWiPPR system.For radar operating angles of incidence of 10, 20 and 30 deg, thecorresponding effective noise temperatures of the system are 1.1×10⁷ K,1.6×10⁶ K and 6.6×10³ K. Each of these three values is substantiallygreater than the system noise temperature T_(system)=300 K indicatingthat the radar system is reverberation limited out to at least an angleof incidence of 30 degrees with respect to the vertical.

FIG. 178 compares the relative energy echo from clear air scatter thathas −30 dBZ reflectivity, The relative echo energy at the receiver isgiven by the radar equation t_(P)P_(echo)/k_(B) where:

${{P_{echo}(\theta)} = {P_{1}\frac{\lambda^{2}G^{2}}{\left( {4\; \pi} \right)^{3}R^{4}}{{dV} \cdot \eta}}},\mspace{14mu} {{dV} = {\frac{4\; \pi}{G}{R^{2}\left( \frac{c}{2B} \right)}}}$

In this equation Ω=4π/G is the solid angle illuminated by the pulse, dVis the volume illuminated by the pulse and η is the volume backscattercoefficient appropriate to −30 dBZ backscatter. The radar horizontalbeamwidth used in estimating backscatter from the surface ϕ_(beam) andthe solid angle Ω are related via ϕ_(beam)≅Ω^(1/2). In MKS units η=10⁻¹⁸K²π⁵λ⁻⁴10^(dBS/10) where the constant K is such that K²≅1 and dBZ=−30 dBin this case.

Detection ranges for radar operation angles of 10°, 20° and 30° areindicated in FIG. 17A. The detection range goes up as the radar beamangle of incidence increases. In general the backscatter from the grounddecays in terms of slant range R like R⁻³ and echoes from clear airscatter decay like R⁻². Thus it is advantageous to operate the radar atlarger values of the radar height z_(r) (thereby increasing slant rangeR) and at larger values of the angle of incidence θ. This strategy willwork so long as the relative clear air echo energy is such thatt_(p)P_(echo)/k₈>10T_(system). The factor of 10 in the foregoingrepresents a margin of safety required to keep false alarms at amanageable level.

In order to assess the effect of backscatter on AWiPPR and optimizeperformance of the system the theoretical backscatter levels in theoperating area need to be related to the noise floor of the radarsystem. This noise floor is composed of components that come from withinthe system and components that arise from outside the system.

It was experimentally observed that the AWiPPR radar could make anaccurate measurement of the dependence of received background noiseenergy as a function of radar tilt angle. This measurement wassubsequently found to be closely related to a theoretical computation ofsky noise temperature using the Blake (1991) standard atmosphere. Theonly difference between the radar measurement and the theoreticalprediction was a linear transformation. This experiment was successfullyrepeated with the radar operating both in active and passive modes. Thenoise floor measurements are shown in FIG. 18.

Mean values of the noise floor at each radar elevation angle areindicated by the larger dots. Individual measurements of the noise floorare indicated by the smaller dots. The measurements show that the radarnoise floor decreases with increasing radar elevation angle.

The noise floor statistic is computed for an individual range-velocitymatrix (RVM) by first removing all Doppler velocity-altitude cells inthe RVM that have signal-to-noise ratio greater than 2 dB. Thiseffectively removes all echoes that are coming from clear air scatter orprecipitation. Next the median level is computed at each altitude,Finally, the noise floor is defined to be the median value of themedians at each altitude.

The noise floor of the radar contains contributions from internalelectronic noise and contributions of noise from the troposphere andcosmos, The sum of the tropospheric noise and noise from the cosmos isknown as sky noise. Sky noise is measured in deg K and it can beconverted to a spectral noise level by multiplying by the Boltzmannconstant k_(B)=1.38 10⁻²³ Joule/° K.. Sky noise is sometimes referred toas brightness temperature since it is measured on a temperature scale.

That portion of the noise energy budget that the radar receives from thetroposphere and beyond is referred to as the brightness temperature(T^(bright)). The relationship between T^(bright) and the verticalprofile of temperature in the atmosphere T(z) is described by theradiative transfer equation shown below (Westwater, 1965). In thisequation α_(v)(z) denotes absorption at frequency v. Although notindicated by the notation in the equation, absorption also depends uponatmospheric temperature, pressure and moisture:

T _(Y) ^(bright) =∫ _(G) ^(∞) T(z)α_(v)(z)exp[−∫₀ ^(z)α_(u)(ζ)dη]dz

FIG. 19 illustrates how the environmentally dependent portion of thisrelationship varies with altitude and frequency. The line in FIG. 19shows the altitude of the 99^(th) percentile of received energy as afunction of frequency, if the columns in the figure are multiplied bytemperature as a function of altitude and then summed down the result isthe atmosphere's contribution to the radar's noise temperature budget.The large region near 60 GHz in the figure is caused by energyabsorption due to the presence of oxygen in the atmosphere. In thisregion absorption is high, radar detection ranges are much shorter thanthey are at 33 GHz.

The following discussion presents an algorithm that relates the noisefloor values observed by the radar to atmospheric brightnesstemperature. In this discussion we will denote the scale constant thatmaps from observed digital radar power in a RVM to the equivalent noisebrightness temperature by α_(scale). The contribution to the radar noisetemperature that comes from internal noise sources measured on a digitalpower scale will be denoted by x_(elec). The constant α_(scale) can befound by solving the following set of equations:

α_(scale)=(x ₉₀ −x _(elec))=T ₉₀,

and

α_(scale)=(xx ₂₅ −x _(elec))=T ₂₅

and where x₂₅ and x₉₀ are the radar noise floor measured in a clear skywith the radar antennas pointed at elevation angles of 25° and 90°. Thenoise values x_(elec), x₂₅ and x₉₀ are measured on the radar digitalscale. The sky noise temperature for the Blake standard atmosphere for aradar operating at 33.4 GHz and pointed in the direction 25 and 90 degare denoted T₂₅ and T₉₀. These two quantities are measured on a Kelvintemperature scale and they have the values T₂₅=38.10 deg K and T₉₀=18.12deg K. The solutions to these two equations are:

${\alpha_{scale} = \frac{T_{25} - T_{90}}{x_{25} - x_{90}}},{and}$${\alpha_{scale}x_{elec}} = \frac{{\alpha_{scale}\left( {x_{25} + x_{90}} \right)} - \left( {T_{25} + T_{90}} \right)}{2}$

where T_(elec)=α_(scale)x_(elec) is the contribution to the radar noisetemperature budget from internal sources including antenna ohmic losses.

The application of this procedure to the noise floor data from is shownin FIG. 20A shows the sensitivity of the noise measurement to radar tiltangle. FIG. 20B shows time variation of sky noise. The data shown wascollected from 1658 to 2103 Z. Sky noise is noise observed by the radarthat comes from the troposphere (0-47 km) and the cosmos (beyond 47 kmfor ordinary radar frequencies). The range of radar elevation angles inFIG. 20B is 25°-90° with 90° corresponding to the vertical (radarlooking straight up). At 33.4 GHz the cosmos contribution to sky noiseis 2.5° K. and is very nearly independent of elevation angle over therange 25°-90°. Thus sky noise in this case is effectively troposphericnoise, (brightness temperature). Smaller points in the plot indicateindividual radar sky noise measurements, Larger points indicate meanvalues. Estimations of the sky noise temperature using algorithms fromBlake (1991) are shown for reference via the dashed line. Computationswith the Blake algorithms for the sky noise temperature have been madeusing on-scene measurements of atmospheric temperature and pressure, Skynoise temperature is lowest near the vertical because at these anglesthe radar beam trajectory spends the least amount of time in the denserportions of the atmosphere that occur at lower altitudes. The constantsand are respectively 0.020 and 111.7° K. The noise floor for the systemon a temperature scale at near vertical elevation angles isapproximately 150° K. of which about 38° K. comes from atmosphericsources.

The flight path of the AWiPPR system 10 and the method thereof duringthe leg 4 data collection is shown in FIG. 21. Specific navigation datarecorded during the flight were mobile airborne platform 12 roll, pitch,yaw and the aircraft vector velocity. Also recorded was the mobileairborne platform 12 altitude. These 7 data types are shown in FIGS.22-23, Also shown in FIG. 23 are slant range to the ground and radarbeam angle of incidence. These last two parameters have been determinedby extracting the position of the maximum of the ground echo in therecorded range velocity matrices. The relationship between slant range Rand aircraft altitude z_(r) is cos ζ=z, 1R. During flight operationsduring leg 4, AWiPPR radar unit 16 was operated at a vertical angle of10 degrees with respect to the normal. This resulted in high backscatterlevels and caused the radar system to saturate, However during the turnat the end of the flight leg, the mobile airborne platform 12 rolled toleft and the radar beams angle of incidence with respect to the groundincreased by about 20°. This reduced backscatter levels and qualityradar data was recorded. A pictorial representation of the turningmaneuver is shown in FIG. 24. The ellipses on the sea level referenceplane indicate the regions where the dominant amount of surface scatteroccurred. The dashed lines in FIG. 24 are the normals to the surface ofthe earth. During the turn shown 20 range velocity matrices wererecorded. Twelve of the range velocity matrices were found to containechoes from the winds in that portion of the convective boundary layerbeneath the aircraft.

Backscattering levels as a function of angle of incidences for theover-water collected data are shown in FIG. 25. Backscattering levelsfor the 12 good files are shown in black. The good files have muchlarger angles of incidence and lower backscatter levels than do theother data files.

FIG. 26 shows the 12 good AWiPPR range velocity matrices as recorded.The ground echo and its side lobe structure are clearly evident, as areboundary layer echo profiles. These profiles represent the Dopplerecho-altitude information from which vector wind velocity as a functionof altitude can be determined.

The estimation of wind vector velocity {v_(x)(z), v_(y)(z), v_(z)(z)}begins with the selection of high quality radar range velocity matrices(data files) that contain Doppler echoes from the wind. The data filesselected for this are shown in FIG. 26. A detailed example of one ofthese range velocity matrices (RVMs) is shown in FIG. 6. In each ofthese there is strong ground echo. Due to the high signal to noiseassociated with this echo, the slant range R_(obx) to the echo and theDoppler velocity of the V_(obx) can be accurately measured. If thevector navigation data is being correctly recorded then it Will be thecase that the computed Doppler velocity of the peak ground echo

V _(com)=η _(beam)(α_(r), β_(p), γ_(y), θ, ϕ)·{u _(x) , u _(y) , u _(z)}

and the observed Doppler velocity are related by the functionalrelationship

mod(V _(com) −V _(max), 2V _(max))−V _(max) =V _(obs)

where mod denotes the modulo function described previously. Thisanalysis was carried out for each of the data files shown in FIG. 26.

The next step in the determination of the vector wind velocity profileis to flip the range velocity matrices upside down and bring the rangeand Doppler velocity of the ground bounce to zero range and zero Dopplervelocity. This accomplished using a combination of circular up-downrolling and circular left right rolling, This can also be doneanalytically as described previously. The results of this procedure areshown in FIG. 27.

The next step in the determination of the vector wind velocity is toextract slant range and observed Doppler vector wind velocity for eachcontact in the RVMs shown in FIG. 27 with signal to noise ratio greaterthan a fixed threshold. Slant range is converted to altitude aboveground level using the corrected navigation data and a combined datafile is produced. A portion of this data file is shown in FIG. 28. Thecolumns in the table are respectively contact altitude, file index,motion compensated Doppler velocity, and the direction cosines of theradar beam at the time of recording with respect to the XYZ coordinatesystem described previously. File numbers are not used in the vectorwind velocity estimation process. They are included in the table onlyfor display and verification purposes. The table shown constitutes theinput to the vector wind velocity estimation algorithm previously.

The outputs of the vector wind velocity estimation process are shown inFIG. 29. The three components of vector wind velocity are represented bythree cubic splines, each with 7 pivot points.

The AWiPPR leg 4 measured winds are compared to the afternoon radiosondevector wind velocity data in FIG. 30. The radiosonde data is indicatedby the dashed lines. The radiosonde does not measure the verticalcomponent of vector wind velocity.

FIG. 31 compares the Doppler velocity profiles from the 12 rangevelocity matrices with clear air contacts to the projection of theestimated vector wind velocity profiles onto the antenna pointingdirection associated with each range velocity matrix. This is a way ofchecking the validity of a prediction in the absence of ground truthdata. The agreement between the measured Doppler data and the projectedDoppler data is quite reasonable in this case.

It has been shown that the AWiPPR system 10 and the method thereof candetermine vector wind velocity by measuring the Doppler velocity ofclear air scatter that is pushed along at the velocity of the wind. Thissame technique can be used when the radar unit 16 detects the motion ofboundary layer particulates such as clouds, rain or snow.

For example, AWiPPR system 10 and the method thereof was operated abovea cloud layer with a nominal elevation of approximately 1900 m aboveground level. From the data collected during this flight operation 12consecutive radar files were selected spanning a time period duringwhich cloud echoes were detected. The radar echo data from the first ofthese files, file number 327 from that day, is depicted in FIG. 32. Thehorizontal axis in the plot is Doppler velocity and the vertical axis isslant range relative to the radar, The radar pulse length during thistime period was T_(m)=190 μs. This corresponds to a maximum positiveunambiguous Doppler velocity of V_(max)=λ/(4T_(m))=11.4 m/s. The upperdashed line in FIG. 32 indicates the position of the ground with respectto the radar. The lower dashed line indicates the position of the cloudformation top with respect to the ground. Individual radar echo data foreach of the 12 files is shown in FIG. 33.

Approximately 20 minutes after the collection of the radar data shown inFIGS. 32 and 33, a balloon-lifted radiosonde was launched from theairport at which the plane took off. The balloon measures horizontalwind speed and direction as a function of altitude as well astemperature, pressure and humidity. Data products based upon the sondemeasurements are depicted in FIGS. 34A-34C, Specifically FIG. 34A showsthe v_(x) and v_(y) components of vector wind velocity. The sonde doesnot measure the v_(z) component of vector wind velocity. FIG. 34B showsa plot of equivalent potential temperature as a function of altitudeabove ground level. Equivalent potential temperature has been computedfrom the sonde measurements of pressure, altitude, temperature andhumidity data collected by the sonde. Equivalent potential temperaturecan be thought of as ordinary potential temperature that has beencorrected for the effects of moisture on air stability. If equivalentpotential temperature has a negative gradient, then the air below isunstable with respect to the air above and will tend to rise. This oftenresults in convection and cloud formation. Note that in FIG. 34C thereis a sharp change in the equivalent potential temperature plot at about1800 m. This feature corresponds to the presence of the cloud layer anda rapid change in the horizontal stratification of the atmosphere.

During the time period in which the radar data shown in FIG. 33 werecollected, the radar was moving along a straight track with a headingvariation of only about 10°. This is not enough beam-pointing directiondiversity to support a full wind speed inversion, but it is possible tocompare the radar Doppler measurements to the sonde vector wind velocitydata projected onto the individual directions of each of the 12 radarbeams. The relationship between Doppler velocity observed by radar unit16, the sonde measurement of vector wind velocity and radar beampointing direction is given by

V _(Doppler)=−{v _(x) ,v _(y) ,v _(z)}η _(beam)

where η _(beam) is the pointing direction of the radar beam at the timeof measurement, {v_(x), v_(y), v_(z)} is the sonde vector wind velocitymeasurement and V_(Doppler) is the Doppler velocity observed by theradar after motion correction. Computations of Doppler velocity profilesbased using each the 12 radar-beam pointing directions are shown in FIG.34C. In making these computations it has been assumed that that thevertical component of wind speed is zero since the sonde does riotmeasure this.

FIG. 35 shows motion corrected radar data compared to estimates ofDoppler velocity based upon the sonde measurements. The lines in FIG. 35are the sonde measurements, They correspond to the Doppler profilesshown in FIG. 34C.

The process of motion correction vertically flips the data Shown in FIG.33 and brings the strong ground bounce echo to zero slant range and zeroDoppler. The radar echoes shown near the ground in FIG. 35 are notassociated with the wind. They are part of the ground echo.

The agreement of the radar data and the sonde data is reasonable. Itmust be kept in mind that the radar and the sonde do not measure exactlythe same thing. Radar unit 16 makes an instantaneous measurement ofDoppler velocity at a fixed-location and as such, it constitutes anEulerian measurement velocity. The sonde infers vector wind velocity bycomputing change of position with respect to time along a drift track.This is a Lagrangian measurement. Additionally, in a cloud bank there islikely to be vertical energy flux. This will be represented in the radarmeasurement but not in the sonde measurement.

The location of the peak ground echo in an AWiPPR range velocity matrixprovides a highly accurate, direct measurement of the ground echoDoppler velocity. If the one or more navigation units 14 are properlyfunctioning then the vector navigation data can be used to predict theground echo Doppler velocity. Substantial average squared differencesbetween the measured and predicted values indicate that there is anerror or bias between the measured and predicted values. Since the radardirectly measures the ground echo Doppler velocity, the most likelyculprit is an error or bias in the navigation system.

The system collected data on 11 flight legs including takeoff andlanding. FIG. 36A compares the navigation estimates of the ground echoDoppler velocity to the directly measured values of the Doppler velocityfor each of the 11 flight legs on that day. If one or more navigationunits 14 are functioning correctly then the data shown should all fallvery closely to the straight line measured navigation. This is clearlynot the case. Data for nine of the flight legs including takeoff andlanding fall below this line. Data for two of the flight legs (Leg08 andLeg10) fall above the line and appear to exhibit anomalous behavior withrespect to the other navigation data.

An improved estimate of the instantaneous six-degree of freedom motionof the aircraft can be obtained by minimizing the squared differencesbetween the directly measured Doppler velocities and the Dopplervelocities predicted by the navigation data. Mathematically the quantitythat is minimized is the sum:

${\chi^{2}({dQ})} = {\frac{1}{\sigma_{Doppler}^{2}}{\sum\limits_{i = 1}^{N}\left\lbrack {V_{meas}^{(i)} - {V_{nav}\left( {Q_{t} + {dQ}} \right)}} \right\rbrack^{2}}}$

where σ_(Doppler) is the error associated with the measurement of theground echo Doppler velocity, N is the number of measurements in thedata sample, V_(meas) ^((i)) is the measured Doppler velocity at timeinstant t_(i), Q_(i)={u_(x) ^((i)), u_(y) ^((i)), u_(z) ^((i)), α^((i)),β^((i)), γ^((i))} is the vector navigation system measurement of theaircraft vector velocity and roll, pitch and yaw. This sum in theforegoing equation is commonly referred to as the chi-squared sum. Thequantity dW={du_(x), du_(y), du_(z), dα, dβ, dγ} is the vector incrementto the navigation data that minimizes the squared-differences betweenmeasurements and predictions of the ground echo Doppler velocity,Procedures for computing the estimates V_(max)(Q_(i)+dQ) from thenavigation data Q, were previously discussed. The chi-square sum X²(dQ)can be minimized by various gradient search techniques (Hughes and Hase,2010). An example of this is shown in FIG. 36B.

In FIG. 36B the data for the takeoff leg have been taken as a proxy forthe non-anomalous navigation data recorded on that day. The takeoff legwas chosen because it spans the full range of ground echo Dopplervelocities observed on that day, The navigation corrections based uponthe minimization of x²(dQ) for the takeoff lekg are shown in the upperleft hand corner of FIG. 36B. As can be seen by examining FIG. 36B, theapplication of this correction to the other eight good legs of dataleads to improved agreement between the measured and predicted groundecho Doppler velocities which in turn will lead to improved vector windvelocity estimates.

In transiting from point P1 to P2, the mobile airborne platform 12undergoes a 90° direction change. The target is located at the origin ofthis rotational coordinate system and the red arrows in FIG. 37Aindicate the direction of fire at the time instants corresponding to P1and P2. The black arrow descending from P1 to the ground track parallelto the instantaneous aircraft direction (as shown by the dashed line)indicates the radar beam direction. At any instance of time this radarbeam looks ahead of mobile airborne platform 12 with the angle ψmeasured with respect to the vertical as shown. In order to achieve adesired look-ahead angle ψ at an aircraft roll angle α, the radar beamwith respect to the mobile airborne platform 12 coordinate system of themobile airborne platform 12 must be steered in the directionsθ_(beam)=ArcSin (cos ψ cos α), ϕ_(beam)=ArcSin(tan α tan θ_(beam)). Fora 40 deg look ahead angle and a −30° roll angle, the two steering beamangles are θ_(beam)=41.6 deg and ϕ_(beam)=−30.8 deg.

The use of a forward-looking radar beam by radar unit 16 may seem an oddchoice. An alternate choice would be to use a radar beam that looks inthe direction of fire. This second choice is not optimal for tworeasons. First, if the radar beam is looking in the direction of fire,then it will potentially receive large echoes from the in-flightprojectiles. These echoes are likely to adversely affect the windmeasurement process of AWiPPR system 10 and the method thereof. Second,the forward looking radar beam measures the cross track component of thewind vector to within a factor of 1/sin ψ at all times provided thatmobile airborne platform 12 is in level flight. Ballistic targeting isan order of magnitude more sensitive to cross-track winds than to alongtrack winds. Thus it is advantageous to continuously monitor thecross-track component of vector wind velocity.

FIGS. 37A-37D show system support to C130 gunship operations. FIG. 37Adepicts the C130 flight path during weapon's fire and the instantaneousdirection of the AWiPPR radar beam. FIG. 37B shows AWiPPR velocity errorsurface for 45° turn. FIG. 37C shows AWiPPR velocity error surface for90° turn. FIG. 37D Cross-track, along-track and up-down velocity errorsfor a range of C130 turn angles.

The turn that the C130 makes during weapons operations provides theangular diversity required to accurately estimate the vector wind fieldfrom the radar Doppler velocity measurements made at different pointsalong the flight path of mobile airborne platform 12. Examples of thisinversion process using measurements recorded by AWiPPR system 10 andthe method thereof during mobile airborne platform 12 turns have beenpreviously given in this document. The point that we focus on here isthe size of the turn that is required to make an accurate vector windvelocity measurement.

In transiting around the fight path circle shown in FIG. 37A, AWiPPRsystem 10 and the method thereof would measure a sequence of Dopplervelocities V _(Doppler)={V₁, V₂, . . . , V_(M)}^(T) at an altitude ofinterest. Radar unit 16 will actually measure Doppler echoes from arange of altitudes but the result that we need here is independent ofthe altitude. These measurements will be made at a sequence of equallyspaced aircraft headings {γ₁, γ₂, . . . , γ_(M)}. The radar beampointing directions at which these measurements are made is given by thevector η _(m) where η _(m)={sin γ_(m) sin ψ, cos γ_(m) sin ψ, −cos ψ}.If A_(radar) is the matrix whose rows contain the radar beam pointingdirections η _(m) then the relationship between the observed Dopplervelocities V _(Doppler) and the vector wind velocity v _(wind)={v_(x),v_(y), v_(z)}^(T) is

V _(Doppler) =−A _(radar) v _(wind)

In writing the above equation it is assumed that the Doppler velocitiesV _(Doppler) have been corrected for the motion of the aircraft. Theleast squares solution for the above equation is

v _(wind)=−(A_(radar) ^(T) A _(radar))⁻¹ A _(radar) ^(T) V _(Doppler)

The error associated with the least squares solution is given by(Clifford, 1973)

$M_{wind} = {{\sigma_{Doppler}^{2}\left( {A_{radar}^{T}A_{radar}} \right)}^{- 1} = {\sigma_{Doppler}^{2}\begin{pmatrix}\sigma_{xx}^{2} & \sigma_{xy}^{2} & \sigma_{xz}^{2} \\\sigma_{xy}^{2} & \sigma_{yy}^{2} & \sigma_{yz}^{2} \\\sigma_{xz}^{2} & \sigma_{yz}^{2} & \sigma_{zz}^{2}\end{pmatrix}}}$

where σ_(Doppler) is the standard deviation of the error associated withthe measurement of Doppler velocity on a radar beam. The diagonalelements of the matrix M_(wind) are the variances of the vector windvelocity measurements in the x(east-west), y(north-south) and z(up-down)directions. The non-diagonal entries in the matrix represent correlatederrors. The error surface associated with the measured data is given by

${{p\left( {x,y} \right)} = {\frac{1}{2\; \pi {C}^{1/2}}{\exp \left\lbrack {{- \frac{1}{2\; \sigma_{doppler}^{2}}}{\overset{\_}{x}}^{T}C^{- 1}\overset{\_}{x}} \right\rbrack}}},\mspace{14mu} {C = \begin{pmatrix}\sigma_{xx}^{2} & \sigma_{xy}^{2} \\\sigma_{xy}^{2} & \sigma_{yy}^{2}\end{pmatrix}}$

where x={x,y}.

FIGS. 378 and 37C show the x (east-west), y (north-south) error surfacefor C130 turns of 45 and 90°. These surfaces have been normalized to apeak value of unity and it has been assumed that σ_(Doppler)=1 m/s. Thetransition into the gray colored background level occurs at −30 dB.There is a significant error spread for the 45° turn, but the 90° turnhas velocity errors which are no greater than about 1 m/s for eithercomponent.

Wind influences the trajectory of a projectile in two primary ways. Itcauses cross-track projectile drift and also influences projectilevertical drop. Unknown drift will cause a left-right miss. Unknown dropwill cause a high-low miss. In order to illustrate the value of AWiPPRvector wind velocity measurements, it is necessary to consider theballistics of a specific projectile and a realistic wind field scenario.Primary weapon systems on C130 gunships include 20 mm, 25 mm and 30 mmcannons. Wind-sensitive ballistic models for these three projectiles arenot readily available. However a wind-sensitive ballistic model for asmaller but similar projectile, the Browning 50 caliber machine gun, isavailable for use on smart phones and computer tablets.

The component of wind that is perpendicular to the firing directionpushes on the side of the projectile and produces cross-track drift. Asimple but accurate model for the cross range drift of a projectile is

x _(drift) =v sin θ(t _(u) −t _(v))

where v is the wind speed in the direction of fire, θ is the winddirection measured clockwise from the direction of fire, t_(o) is theactual time of flight of the projectile to the target and t_(v) is thetheoretical time of flight of the projectile in a vacuum. Cross-trackdrift is maximized when the wind angle θ is either 90° or 270°. Thismodel is an adaptation of Herbert A. Leupold's equation (Leopold, 1996).It produces results that are in agreement with the Ballistics for iPadapplication described in Zdziarski (2016).

For a Browning 50 caliber machine gun using a 750 grain projectile thatis fired level with a 2700 m/s muzzle velocity, the actual time offlight of the projectile to a range of 2000 yd is 3.22 sec and thetheoretical time of flight in a vacuum to this distance is 2.22 sec. Thevertical drop for this projectile at a range of 2000 yd is given by theequation

$z_{drop} = {1600 + {34\frac{v}{10}\cos \; \theta}}$

where z_(drop) is measured in inches and the wind speed v is measured inm/s. Vertical drop is maximal when the projectile is fired into the wind(θ=0 deg) and is minimal when the wind is behind the bullet (θ=180 deg).

In order to illustrate the importance of wind field knowledge onprojectile targeting two cases are considered: 1) An unknown wind fieldwith random direction over the angular range 0° to 360° and wind speedin the range 0-20 m/s. 2) A ground truth wind field with v=10 m/s andθ=70 deg. AWiPPR system 10 and the method thereof is assumed to measurethis wind ground truth wind field to within 1 m/s in speed and 6 deg inazimuth. FIGS. 38A and 37B illustrate how targeting accuracy improves asthe uncertainty in wind speed and direction decreases. The verticalarrows in FIGS. 38A-38B represent the state of exact knowledge of thewind field. If the wind field is known exactly, then cross-track driftand along-track drop can be exactly calculated. For the case ofuncertain wind, probability distributions are used to representuncertainty in the effects of wind on targeting. These probabilitydistributions have been computed using a simple Monte Carlo model andthe two previously presented equations describing drift and drop for theBrowning 50 caliber machine gun projectile. Referring to FIG. 38A, acomplete lack of knowledge of wind speed and direction results inprojectile drifts over the range 0±116.3 m The use of AWiPPR wind speedmeasurements reduces this drift to the range 9.3±2.0 m. The effects ofwind on projectile drop are shown in FIG. 38B. In FIG. 38B an additionalground truth scenario in which the wind direction is changed from 70°deg to 170° is presented. The use of AWiPPR wind estimates again resultsin reduced error spreads and improved targeting. Referring to FIG. 378,when the wind is at 70°, azimuth wind effects lengthen flight time andproduce more drop. When the wind is at 170°, flight time to the targetis reduced and there is less drop.

FIG. 39 shows one example of the primary steps performed by AWiPPRsystem 10 and the method thereof. In this example, a signal istransmitted, step 200. A return signal from the CAS of volumetrictargets is received, step 202. The signal is processed to extract arange and velocity, step 204. Navigational and IMU data are provided,step 206. The navigation data and IMU data are time aligned, step 208.Extracted range and velocity and time aligned data are used to determineantenna pointing direction and velocity of the mobile platform, step210. A list of targets (including measured range to target, measuredvelocity, antenna positioning direction, and velocity of the mobileplatform) is extracted from a list of targets, step 212. The ground echois then located, step 214. Using the location of the ground echo, avelocity correction is determined, step 216. The target velocity iscorrected using the velocity correction, step 218. The coordinate systemfor the target range is transformed to an earth-centered coordinatesystem, step 220. A three-dimensional wind vector is represented as aset of three coupled natural splines, step 222. The spine ordinatepoints are determined by requiring equal information in each span, step224. The unknown scale or spline values that fit the data in a leastsquares sense are determined, step 226. The Chi-square sum between themeasured wind data and estimated wind is minimized, step 228. The finalwind measurement is then generated, step 230.

As disclosed herein, one or more navigation units 14, the variouscomponents of radar unit 16, and IMU 38, and field control system 62 mayinclude one or more processors, an application-specific integratedcircuit (ASIC), firmware, hardware, and/or software (including firmware,resident software, micro-code, and the like) or a combination of bothhardware and software. Computer program code for the programs forcarrying out the instructions or operation of one or more units 14,radar unit 16, and IMU 38, and field control system 62 discussed abovewith reference to one or more of FIGS. 1-38 may be written in anycombination of one or more programming languages, including an objectoriented programming language, e.g., C++, Smalltalk, Java, and the like,or conventional procedural programming languages, such as the “C”programming language or similar programming languages.

Although specific features of the invention are shown in some drawingsand not in others, this is for convenience only as each feature may becombined with any or all of the other features in accordance with theinvention, The words “including”, “comprising”, “having”, and “with” asused herein are to be interpreted broadly and comprehensively and arenot limited to any physical interconnection. Moreover, any embodimentsdisclosed in the subject application are not to be taken as the onlypossible embodiments. Other embodiments will occur to those skilled inthe art.

In addition, any amendment presented during the prosecution of thepatent application for this patent is not a disclaimer of any claimelement presented in the application as filed: those skilled in the artcannot reasonably be expected to draft a claim that would literallyencompass all possible equivalents, many equivalents will beunforeseeable at the time of the amendment and are beyond a fairinterpretation of what is to be surrendered (if anything), the rationaleunderlying the amendment may hear no more than a tangential relation tomany equivalents, and/or there are many other reasons the applicantcannot be expected to describe certain insubstantial substitutes for anyclaim element amended.

Other embodiments will occur to those skilled in the art and are withinthe following claims.

What is claimed is:
 1. An airborne wind profiling portable radar (AWiPPR) system comprising: a mobile airborne platform including one ormore navigation units configured to produce navigation data including atleast the position and orientation of the mobile airborne platform; aradar unit mounted and positioned to the mobile airborne platform, theradar unit configured to transmit a wide-band frequency modulatedcontinuous wave radar signal in a downward direction from the mobileairborne platform towards the ground and configured to continuouslyreceive a reflected signal from each of a plurality of clear airscatters (CAS) targets or volumetric targets and output radar data; aninertial measurement unit (IMU) in communication with the one or morenavigation units and the radar unit configured to receive the navigationdata and determine the position and orientation of the radar at aspecific point in time and output IMU data; a data acquisition unit incommunication with the radar unit and the configured to receive and timealign radar data and the IMU data for each reflected signal from each ofthe plurality of CAS targets or volumetric targets to provide an antennapointing direction for each received reflected signal; wherein the dataacquisition unit is configured to process the time aligned radar dataand IMU data to determine a distance and a Doppler velocity of each ofthe plurality of CAS targets or volumetric targets, provide a range, avelocity, and an antenna pointing direction for each of the plurality ofCAS targets or volumetric targets, and calculate a vector wind velocityusing the range, the velocity, and the antenna pointing direction foreach of the plurality of CAS targets or volumetric scatters targets; andwherein the data acquisition unit is configured to further correct therange, the velocity, and/or the antenna pointing direction of each ofthe plurality of CAS targets or volumetric targets to accommodate for amotion shift in data produced by one or more of: a relative motion andorientation of the mobile airborne platform, a Doppler spread in therange, the velocity and/or the antenna pointing direction, and a groundecho.
 2. The system of claim 1 in which a motion shift in the range, thevelocity, and the antenna pointing direction does not include a Dopplerwrap and wherein the navigation unit is configured to generate anavigation correction for each reflected signal by adding a speed of themobile airborne platform provided by the one or more navigation units tothe determined Doppler velocity for each of the plurality of CAS targetsor volumetric targets and rotating the range, the velocity, and theantenna pointing direction into a coordinate system centered beneath themobile airborne platform.
 3. The system of claim 1 wherein the dataacquisition unit is responsive to sparseness of the plurality of CAStargets, shifts in the position of the mobile airborne platform over apredetermined measurement window, and navigation correction applied to aset of reflected signals from the plurality of CAS targets or volumetrictargets and the data acquisition unit is configured to infer a Dopplerfield vector for each of the plurality of CAS targets or volumetrictargets as a set of three coupled cubic splines derived from themeasured Doppler velocity data for the plurality of CAS targets orvolumetric using a non-parametric function estimation.
 4. The system ofclaim 3 wherein the data acquisition unit is configured to generate avector wind field from the set of three coupled cubic splines by:representing the second derivative of the cubic splines as a piecewisecontinuous linear function (f(z)); integrating the function twice toyield a cubic polynomial producing a plurality of pivot points of thecubic splines, wherein the function (f(z)) must pass through the pivotpoints and be zero at the first and last pivot points such that thecubic splines are natural splines; determining a plurality of unknownspline ordinate points from the altitude and velocity data obtained bythe data acquisition unit, wherein a minimum spline abscissa value isequal to the minimum altitude and velocity data values, and wherein amaximum spline abscissa value is equal to the maximum altitude andvelocity data values, wherein the abscissa of the altitude and velocitydata lies in an abscissa interval of the cubic splines, and wherein theordinate points represent the velocity of the unknown wind field;wherein such the abscissa intervals are determined by ensuring that allabscissa intervals contain equal amounts of information; and whereinsuch that the relationship between the observed velocity data and thecubic splines is given by:V _(N) _(d) _(xi) =A _(N) _(d) _(×3N) f _(3N) where V is the vector ofthe obtained velocity data, N is the number of data points, f is thevector of a set of cubic spline coefficients, A is an informationmatrix, and Af is a cubic spline model.
 5. The wind-profiling radarsystem of claim 4 wherein the data acquisition unit further fits thecubic spline model Af to the obtained velocity data V using aleast-squares technique.
 6. The wind-profiling radar system of claim 4wherein the data acquisition unit further minimizes a difference betweenthe obtained velocity data V and the cubic spline model Af by obtaininga maximum likelihood estimate of f.
 7. The wind-profiling radar systemof claim 1 wherein the data acquisition unit determines a requiredminimum slant distance of the radar unit relative to the ground from thereflected signal that yields a maximum allowable return signal into theradar unit before the performance of the radar unit is reduced tosaturation or compression.
 8. The wind-profiling radar system of claim 7wherein the data acquisition unit is configured to determine therequired incidence angle using a Beckman and Spizzichino model.
 9. Thewind-profiling radar system of claim 7 wherein a pointing angle of theradar unit relative to the mobile airborne platform is adjustable andwherein the radar unit adjusts the pointing angle of the radar unitbased on the determined required incidence angle.
 10. The wind-profilingradar system of claim 9 wherein the pointing angle of the radar unit ispointed at an angle relative to a normal to the ground of greater thanabout 0° and less than about 90°.
 11. The wind-profiling radar system ofclaim 1 wherein the data acquisition unit is further configured toestimate the wind vector velocity by: selecting a plurality ofmeasurements containing a CAS target or volumetric target anddetermining a slant distance and Doppler velocity of a ground echo fromeach; performing the required coordinate transformations such that thedistance and Doppler velocity of the ground echo are at zero distanceand velocity; and extracting a slant distance and Doppler vector windvelocity for each of the CAS targets or volumetric targets in theplurality of measurements above a fixed signal-to-noise threshold; andconverting the slant distance to an altitude above ground level usingthe navigation data from the one or more navigation units.
 12. Thewind-profiling radar system of claim 11 wherein the data acquisitionunit is further configured to minimize the chi-square sum between themeasured wind vector velocities and the estimated wind vector velocitiesby a gradient search technique.
 13. The wind-profiling radar system ofclaim 1 wherein the radar unit transmits with a sweep width configuredto match the back-scattering characteristics of the plurality of CAStargets or volumetric targets.
 14. The wind-profiling radar system ofclaim 13 wherein the sweep widths range from about 6 MHz to about 200MHz.
 15. The wind-profiling radar system of claim 1 wherein the radarunit transmits in a waveform selected from one or more of: linearfrequency modulated (FM) waveform, a phase coded waveform, or non-linearFM waveform.
 16. The wind-profiling radar system of claim 1 wherein theradar unit is configured to transmit at a carrier frequency in the Kaband.
 17. The wind-profiling radar system of claim 1 wherein the radarunit is configured to convert the wide-band frequency modulatedcontinuous wave radar signal to a Ka band and filter and amplify the Kaband signal prior to transmission thereof.
 18. The wind-profiling radarsystem of claim 1 wherein the radar unit is configured to receive thereflected signal from each of the plurality of CAS targets or volumetrictargets, amplify the received signal, down-convert the received signalto a baseband received signal, and filter and amplify the receivedsignal.
 19. The wind-profiling radar system of claim 18 wherein thedown-conversion is homodyne single side band.
 20. The wind-profilingradar system of claim 18 wherein the down-conversion is homodyne and isdual side band.
 21. The wind-profiling radar system of claim 1 whereinthe radar unit includes one or more antennas.
 22. A method ofdetermining a vector wind velocity as a function of altitude above theground on a mobile airborne platform, the method comprising: providingnavigation data including at least positioning and orientation of themobile airborne platform; transmitting a wide band frequency modulatedcontinuous wave radar signal in a downward direction from the mobileairborne platform towards the ground; continuously receiving a reflectedsignal from each of a plurality of clean air scatter (CAS) targets orvolumetric targets and outputting radar data; determining the positionand orientation of a radar unit mounted and positioned on the mobileairborne platform at a specific point in time and outputting positionand orientation data; time aligning the radar data with the position andorientation data for each reflected signal from each of the plurality ofCAS targets or volumetric targets to provide an antenna pointingdirection for each of the plurality of CAS targets or volumetricscatters targets; processing the timed aligned radar data and positionand orientation data to determine a distance and Doppler velocity foreach of the plurality of CAS targets or volumetric targets and provide arange, a velocity, and an antenna pointing direction for each of theplurality of CAS targets or volumetric targets and calculating a vectorwind velocity using the range, the velocity, and the antenna pointingdirection for each of the plurality of CAS targets or volumetrictargets; and further correcting the range, the velocity, and/or theantenna pointing direction of each of the plurality of CAS targets orvolumetric targets to accommodate for a motion shift in the dataproduced by one or more of: a relative motion in orientation of themobile airborne platform, a Doppler spread in the range, the velocity,and/or the antenna pointing direction and a ground echo.
 23. The methodof claim 22 in which a shift in the range, the velocity, and the antennapointing direction does not include a Doppler wrap and further includinggenerating a navigation correction for each reflected signal by adding aspeed of the mobile airborne platform to the determined Doppler velocityfor each of the plurality of CAS targets or volumetric targets androtating the range, the velocity, and the antenna pointing directioninto a coordinate system centered beneath the airborne platform.
 24. Themethod of claim 22 further including detecting sparseness of theplurality of CAS targets, shifts in position of the mobile airborneplatform over a predetermined measurement window and navigationcorrection applied to a set of reflected signals from the plurality ofCAS targets and inferring a Doppler field vector for each of theplurality of CAS targets or volumetric targets as a set of three cubicsplines derived from the measured Doppler velocity data for theplurality of CAS targets or volumetric scatters targets using anon-parametric function estimation.
 25. The method of claim 24 furtherincluding generating a vector field from each of a set of cubic splinesby: representing the second derivative of the cubic splines as apiecewise continuous linear function (f(z)); integrating the functiontwice to yield a cubic polynomial producing a plurality of pivot pointsof the cubic splines, wherein the function (f(z)) must pass through thepivot points and be zero at the first and last pivot points such thatthe cubic splines are natural splines; determining a plurality ofunknown spline ordinate points from the altitude and velocity dataobtained by the data acquisition unit, wherein a minimum spline abscissavalue is equal to the minimum altitude and velocity data values, andwherein a maximum spline abscissa value is equal to the maximum altitudeand velocity data values, wherein the abscissa of the altitude andvelocity data lies in an abscissa interval of the cubic splines, andwherein the ordinate points represent the velocity of the unknown windfield; wherein such that the abscissa intervals are determined byensuring that all abscissa intervals contain equal amounts ofinformation; wherein such that the relationship between the observedvelocity data and the cubic splines is given by:V _(N) _(d) _(×i) =A _(N) _(d) _(×3N) f _(3N) where V is the vector ofthe obtained velocity data, N is the number of data points, f is thevector of a set of cubic spline coefficients, A is an informationmatrix, and Af is a cubic spline model.
 26. The method of claim 25further including fitting the cubic spline model of Af to obtain thevelocity data using a least squares technique.
 27. The method of claim24 further including minimizing the difference between the obtainedvelocity data V and the cubic spline model Af by obtaining a maximumlikelihood estimate of f.
 28. The method of claim 22 further includingdetermining a required minimum slant distance of a radar unit disposedon the mobile airborne unit relative to the ground from the reflectedsignal that yields a maximum allowable signal return before aperformance of the radar unit is reduced to saturation or compression.29. The method of claim 28 further including determining the requiredincidence angle using a Beckman and Spizzichino model.
 30. The method ofclaim 27 further including providing a pointing angle of the radar unitrelative to the mobile airborne platform that is adjustable and theradar unit adjusts a pointing angle of the radar unit based on thedetermined required incidence angle.
 31. The method of claim 30 whereinthe pointing angle of the radar unit is pointed at an angle relative toa normal to the ground of greater than about 0° and less than about 90°.32. The method of claim 22 further including estimating the vector windvelocity by: selecting a plurality of measurements containing a CAStarget or volumetric scatter target and determining a slant distance andDoppler velocity of a ground echo from each; performing the requiredcoordinate transformations such that the distance and Doppler velocityof the ground echo are at zero distance and velocity; extracting a slantdistance and Doppler vector wind velocity for each of the CAS targets inthe plurality of measurements above a fixed signal-to-noise threshold;and converting the slant distance to an altitude above ground levelusing the navigation data from the one or more navigation units.
 33. Themethod of claim 32 further including minimizing the Chi-square sumbetween the measured wind vector velocity and the estimated wind vectorvelocity by a gradient search technique.
 34. The method of claim 22 inwhich the wide-band frequency modulated continuous wave radar signaltransmits with a sweep width configured to match the back-scatteringcharacteristics of the CAS targets or volumetric targets.
 35. The methodof claim 34 in which the sweep widths are in the range of about 6 MHz toabout 200 MHz.
 36. The method of claim 22 in which the wide-bandfrequency modulated continuous wave radar signal includes one or more oflinear frequency modulated (FM) waveform, a phase coded waveform, ornon-linear (FM) waveform.
 37. The method of claim 22 in which thewide-band frequency modulated continuous wave radar signal transmits acarrier frequency in the Ka band.
 38. The method of claim 22 furtherincluding converting the wide-band frequency modulator continuous waveradar signal to a Ka band, filtering and amplifying the Ka band prior totransmission thereof.
 39. The method of claim 22 further includingreceiving the reflected signal from each of the plurality of CAS targetsor volumetric scatters targets, amplifying the received signal, downconverting the received signal to a base band received signal andfiltering and amplifying the received signal.
 40. The method of claim 39wherein the down conversion is a homodyne single side band.
 41. Themethod of claim 38 wherein the down conversion is a homodyne and is adual side hand.